Third-Order Inviscid and Second-Order Hyperbolic

Navier-Stokes Solvers for Three-Dimensional Inviscid

and Viscous Flows

Yi Liu∗and Hiroaki Nishikawa†

National Institute of Aerospace, Hampton, VA 23666

This paper presents third-order-inviscid implicit edge-based solvers for three-dimensionalinviscid and viscous flows on unstructured tetrahedral grids. Third-order edge-based schemehas been implemented into NASA’s FUN3D code for inviscid terms. Second-order edge-basedhyperbolic Navier-Stokes schemes, which achieve third-order accuracy in the inviscid terms,have also been implemented. Some key improvements are reported for the hyperbolic Navier-Stokes schemes. Third-order accuracy is verified by the method of manufactured solutionsfor unstructured tetrahedral grids. Developed schemes are compared for some representativetest cases for three-dimensional inviscid and viscous flows.

I. Introduction

Edge-based discretization has been widely used in practical Computational Fluid Dynamics (CFD) solvers[1, 2, 3, 4, 5, 6, 7] for inviscid and viscous flow applications. Refs.[8, 9] show that the node-centered edge-baseddiscretization achieves third-order accuracy for hyperbolic systems on arbitrary simplex-element grids, trianglesin two dimensions (2D) and tetrahedra in three dimensions (3D) with quadratic least-squares (LSQ) methodsand linearly-extrapolated fluxes, and third-order accuracy has been confirmed later in Ref.[10]. The third-orderedge-based discretization is highly economical in that the residual is computed over a single loop over edgeswith a single numerical flux evaluation per edge. Moreover, it delivers third-order accurate solutions on lineartetrahedral grids even for curved geometries [11]. Ref.[12] extended the third-order scheme to the viscous termsin 2D, and Refs.[13,14] extended it to 3D prismatic strand grids by combining it with a high-order finite-differencescheme applied in the direction normal to a surface. Our interest, on the other hand, is to extend the third-order scheme to Navier-Stokes formulations on purely tetrahedral grids for improving practical unstructured-grid turbulent-flow solvers, allowing accurate and robust simulations with arbitrary isotropic/anisotropic gridadaptation for complex geometries [15, 16, 17, 18, 19]. This paper reports progress in the development of third-order edge-based schemes for 3D turbulent flows, culminating an effort published in a series of papers [11,20,21,22,23,24]. The schemes are implemented in the framework of NASA’s FUN3D code [1], which is a well-validated3D unstructured-grid solver developed by NASA Langley Research Center.

In this paper, we focus on Navier-Stokes formulations and consider two approaches. The first approach isa straightforward application of the third-order edge-based scheme to the inviscid terms using quadratic LSQgradient reconstruction and linearly-extrapolated inviscid fluxes. The default FUN3D viscous scheme (i.e., theP1 continuous Galerkin discretization) is added to the inviscid scheme. The second approach is the hyperbolicNavier-Stokes (HNS) method [23, 24, 25, 26], which reformulates the viscous terms as a hyperbolic system withthe solution gradients introduced as additional unknowns. In the HNS method, third-order accuracy in theinviscid terms can be achieved without quadratic LSQ methods [24] because second-order accurate gradientsare directly obtained as unknowns. The HNS schemes have been demonstrated for 3D flows in Ref.[26]. Thispaper introduces some key improvements to the methodologies presented in Ref.[26]. Both approaches areimplemented in FUN3D, verified, and applied for inviscid and viscous flow problems.

It should be noted that the relative merits of the two approaches are not immediately clear. Although theHNS schemes have an advantage of achieving third-order accuracy in the inviscid terms without the discretization

∗Senior Research Scientist (yi.liu@nianet.org), National Institute of Aerospace, 100 Exploration Way, Hampton, VA 23666 USA,Senior Member AIAA

†Associate Research Fellow (hiro@nianet.org), National Institute of Aerospace, 100 Exploration Way, Hampton, VA 23666 USA,Associate Fellow AIAA

1 of 41

American Institute of Aeronautics and Astronautics

Dow

nloa

ded

by H

iroa

ki N

ishi

kaw

a on

Jun

e 10

, 201

6 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.2

016-

3969

46th AIAA Fluid Dynamics Conference

13-17 June 2016, Washington, D.C.

AIAA 2016-3969

Copyright © 2016 by Hiroaki Nishikawa, Yi Liu. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

AIAA Aviation

stencil extension required for a quadratic LSQ fit, the schemes use extra variables to form a hyperbolic viscoussystem. The combination of the inviscid third-order scheme and the Galerkin viscous discretization does notrequire additional variables and equations. However, it requires the discretization stencil extension beyondneighbors of the neighbors and quadratic LSQ gradients that may not be robust on practical 3D unstructuredviscous grids. Another important aspect is that the HNS schemes are known to produce high-order and high-quality gradients (e.g., viscous stresses, heat fluxes, and vorticity) on irregular unstructured grids [23,24,26] whilethe first approach can produce only first-order accurate gradients (unless the inviscid terms dominate) pollutedby numerical noise on irregular grids. Moreover, the HNS solvers are known to eliminate numerical stiffnessarising from second derivatives in the governing equations and converge faster on fine grids than conventionalsolvers [23, 24, 25, 26]. Efficiency comparison is, thus, not a simple matter, and will be even more complicatedif other aspects, such as solver constructions, code optimizations, other discretization alternatives, etc., arebrought into discussion. For example, if one focuses on gradient accuracy (e.g., viscous drag), the complexityand efficiency of the HNS schemes need to be compared with those of third-order methods such as P2 continuousor discontinuous Galerkin schemes. For these reasons, this paper focuses on accuracy and robustness ratherthan computational cost and efficiency, towards third-order edge-based unstructured-grid solvers that can solveany problem that the current state-of-the-art second-order solver can solve.

There exist other techniques for improving accuracy of node-centered schemes for the inviscid terms. Thesemethods [27,28] are based on high-order reconstructions and fundamentally different from the third-order methodconsidered here. The methods are not truly high order on general unstructured grids because the flux integralis approximated by a low-order quadrature formula. The third-order edge-based scheme discussed in thispaper achieves third-order accuracy on general simplex-element grids without high-order quadrature due to theexactness of quadratic flux integration on simplex-element grids. Also, a subtle but important difference is thatthe schemes [27, 28] attempt to directly reduce (if not eliminate) second-order truncation errors of the inviscidapproximation whereas the edge-based scheme generates a second-order truncation error containing derivativesof the inviscid terms, which thus vanish on exact solutions or equivalently at a zero residual limit [11,20,22]. Theproperty of achieving high-order accuracy by vanishing residuals, often called the residual property, is common toeconomical high-order methods such as the residual-compact method [29] and the residual-distribution method[30]. The residual property allows constructions of high-order schemes with a relatively compact stencil, butit requires compatible discretizations for inviscid, viscous, and source terms to preserve the design accuracy[20,30].

The paper is organized as follows. In Section II, the governing equations are described. In Section III, thenode-centered edge-based discretization is described, and an improved HNS scheme is presented. In Section IV,a general boundary flux quadrature and boundary conditions are discussed. In Section V, numerical results arepresented. Finally, Section VI concludes the paper with remarks.

II. Governing Equations

II.A. Form of Governing Equations

We consider governing equations of the form:

P−1∂τU+ divF = S, (II.1)

where τ is a pseudo time, P is a local-preconditioning matrix (which is a simple diagonal scaling matrix), Uis a solution vector, F is a flux tensor, and S is a source term. In this paper, we focus on steady problems,and therefore seek steady solutions in the limit of the pseudo time. Unsteady computations performed byimplicit time-stepping schemes will be reported in a subsequent paper. For inviscid flow problems, the governingequations are the Euler equations with P is the identity matrix, U is a vector of conservative variables, F = Fi

is the inviscid flux tensor, and S = 0 except in the cases of manufactured solutions, where source terms aregenerated to enforce a given exact solution. For viscous flow problems, the governing equations are the Navier-Stokes equations, where the flux tensor consists of the inviscid and viscous parts: F = Fi + Fv. In the case ofthe HNS equations, we employ the HNS20 system [24,26], where U has extra variables as mentioned below, P isa diagonal matrix that scales the equations for the extra variables, and S contains source terms associated withthe hyperbolic formulation. The extra variables are called the gradient variables, and they are proportional tothe gradients of the primitive variables when the pseudo-time terms vanish or ignored:

r = νρ ∇ρ, g = [gu,gv,gw] = [µv∇u, µv∇v, µv∇w], q = − µh

γ(γ − 1)∇T, (II.2)

2 of 41

American Institute of Aeronautics and Astronautics

Dow

nloa

ded

by H

iroa

ki N

ishi

kaw

a on

Jun

e 10

, 201

6 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.2

016-

3969

where ρ is the density, v = (u, v, w) is the velocity vector, T is the temperature, γ is the ratio of specific heats,and

νρ = Vmin, µv =4µ

3, µh =

γµ

Pr, (II.3)

Vmin is the minimum dual control volume of a given grid [26] (i.e., νρ is a global constant), µ is the viscositygiven by Sutherland’s law, and Pr is the Prandtl number. See Refs.[24, 26] for more details of the HNS20system. All solutions are non-dimensionalized by free-stream values except that the velocity and the pressureare normalized by the free-stream speed of sound and the free-stream dynamic pressure, respectively [1,31,26].

II.B. Length Scale for HNS

The HNS20 system has relaxation-time parameters [23,24,25,26]:

Tρ =L2

νρ, Tv =

L2

νv, Th =

L2

νh, (II.4)

where νv = µv/ρ, νh = µh/ρ, and L is a length scale defined, in the previous studies, as

L =1

2π. (II.5)

It has been pointed out [32,33] that the length scale must be reduced for high-Reynolds-number boundary-layerflows, or iterative solvers may diverge. The following modified length scale effectively resolves the issue in manycases:

L =Lr

1/2 +√

ReLr + 1/4, (II.6)

where Lr is the optimal length scale derived for the advection-diffusion equation in Ref.[34], and ReLr is the freestream Reynolds number based on Lr. The modified length scale is applied only to Tv for problems presentedin this paper. Whether the above formula is the optimal one remains an open question. A complete account ofhigh-Reynolds-number issues will be given in a separate paper.

III. Node-Centered Edge-Based Scheme

III.A. Discretization

The governing equations are discretized on a tetrahedral grid by the node-centered edge-based method. Thediscretization at a node j is defined as

VjdUj

dτ= −Pj

∑k∈{kj}

ΦjkAjk − SjVj

, (III.1)

where Pj is the diagonal scaling matrix, Vj is the measure of the dual control volume around, {kj} is a setof neighbors of the node j, Φjk is a numerical flux, and Ajk is the magnitude of the directed area vector,which is a sum of the directed-areas corresponding to the dual-triangular faces associated with all tetrahedralelements sharing the edge [j, k] (see Figure 1). The numerical flux is computed at the edge midpoint. The linearreconstruction is performed with solution gradients computed by a linear/quadratic LSQ fit.

In the HNS method, the edge-based dfiscretization is applied to both the inviscid and viscous terms. Twotypes of HNS20 schemes are considered: HNS20-I and HNS20-II [23, 24, 26]. The former corresponds to theedge-based discretization directly applied to the HNS20 system, and the latter is obtained from HNS20-I byreplacing LSQ gradients of the primitive variables by the gradient variables. The HNS20-II scheme has anadvantage of achieving third-order accuracy in the inviscid limit [24]. In this paper, we introduce an improvedversion of the HNS20-I scheme, designated as HNS20-I(Q), which can also achieve third-order accuracy in theinviscid limit. The details of HNS20-I(Q) will be given in Section III.E. For conventional schemes, we considerthe default FUN3D inviscid scheme and the third-order inviscid scheme, which are designated as FUN3D andFUN3D-i3rd, respectively. In both FUN3D and FUN3D-i3rd, the Galerkin discretization of the viscous terms,

3 of 41

American Institute of Aeronautics and Astronautics

Dow

nloa

ded

by H

iroa

ki N

ishi

kaw

a on

Jun

e 10

, 201

6 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.2

016-

3969

Scheme

Discretization Jacobian

Inviscid Viscous Inviscid Viscous

Flux LSQ (ρ,v, p) Flux LSQ (r,g,q)

FUN3D Roe(2nd) : 2 Linear Galerkin(2nd) : 1 None Van Leer Exact

FUN3D-i3rd Roe(3rd) : 3 Quadratic Galerkin(2nd) : 1 None Van Leer Exact

HNS20-I Roe(2nd) : 2 Linear Upwind(2nd) : 2 Linear Van Leer Upwind

HNS20-I(Q) Roe(3rd) : 2 C-quadratic Upwind(2nd) : 2 Linear Van Leer Upwind

HNS20-II Roe(3rd) : 2 None Upwind(2nd) : 2 Linear Van Leer Upwind

Table 1: Summary of discretizations and Jacobians. The number on the right side of each colon indicates thelevel of neighbors contributing to the residual at a node: 1 = up to the neighbors, 2 = up to neighbors of theneighbors, 3 = up to neighbors of the neighbors of the neighbors. Order of accuracy is indicated by 2nd and3rd in the parentheses.

which can be implemented as a loop over edges for tetrahedral grids, is added for viscous flow problems. Thesediscretizations are summarized in Table 1.

The discretization yields a global system of residual equations, which is solved by an implicit defect-correctionsolver similar to the one described specifically for the HNS schemes in Ref.[26]. The linear relaxation is performedby a multi-color Gauss-Seidel scheme available in FUN3D. In this study, we perform 15 linear relaxations perimplicit iteration for all cases. The residual Jacobians are constructed as summarized in Table 1. The Jacobianmatrix is constructed exactly for the Galerkin viscous discretization, and approximately for other discretizationsby the linearization of first-order-accurate residuals. The inviscid Jacobian is constructed by the linearizationof Van Leer’s flux vector splitting scheme. The Jacobian consists of 5×5 blocks for FUN3D and FUN3D-i3rd,and 20×20 blocks for HNS20-I, HNS20-I(Q), and HNS20-II. The Jacobian is updated based on an efficientalgorithm available in FUN3D, which automatically adjusts the Jacobian-update-frequency based on residualreduction. The pseudo-time term in Equation (III.1) is discretized by the first-order backward Euler scheme,and the pseudo time step is defined locally with CFL = 200 for all cases unless otherwise stated. In future, theimplicit defect-correction solver will be incorporated, as a variable preconditioner, into a Newton-Krylov solver[35,36].

III.B. Third-Order Accuracy in the Inviscid Terms

The edge-based quadrature used in the discretization (III.1) is known to be exact for linearly-varying fluxeson arbitrary simplex-element (triangular/tetrahedral) grids and thus sufficient for second-order accuracy on suchgrids. However, it is formally first-order accurate on other types of elements unless certain regularity conditionsare satisfied (see Ref.[37] and Appendix E in Ref.[38]). A recent study [8, 9] revealed that the discretization(III.1) can achieve third-order spatial accuracy (in the case of ∂U/∂τ = 0 and S = 0) on arbitrary simplex-element grids with two modifications: quadratic LSQ instead of linear LSQ for nodal gradients, and linearlyreconstructed fluxes instead of fluxes evaluated from linearly reconstructed solutions. A distinguished featureof the third-order edge-based discretization is that it does not require high-order quadrature formulas. Theresidual is computed in a loop over edges with a single numerical flux evaluation per edge, similar to the second-order discretization. This edge-based discretization is different from a typical (not edge-based) finite-volumediscretizations. Figure 2 illustrates the difference on a triangular grid. In a typical finite-volume method, theflux integration is performed over the dual-control volume boundary with a quadrature rule applied to eachstraight boundary piece. With this midpoint quadrature, where fluxes are evaluated at the midpoint of the dualinterval as illustrated in Figure 2(a), the finite-volume method is second-order accurate for any type of element.In the edge-based method, the flux integration uses a flux evaluated at the edge midpoint, as illustrated in Figure2(b). The edge-based quadrature is more economical, with a significantly fewer flux evaluations especially in3D [39]. For third-order accuracy, a typical finite-volume method requires a high-order quadrature formulawith at least two quadrature points per straight dual interval, as illustrated in Figure 2(c). In 3D, three points

4 of 41

American Institute of Aeronautics and Astronautics

Dow

nloa

ded

by H

iroa

ki N

ishi

kaw

a on

Jun

e 10

, 201

6 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.2

016-

3969

per triangular dual face (as shown in Figure 1(b)) will be required [40]. The edge-based discretization (III.1)can achieve third-order accuracy with exactly the same edge-based quadrature (see Figure 2(d)) on simplex-element grids in both 2D and 3D. It is also important to note that the third-order edge-based discretization isa special feature relying on the combination of the edge-based quadrature and the linear extrapolation. Whilea third-order finite-volume method requires a quadratic extrapolation to obtain the solution and the flux at aquadrature point, the edge-based method can be third order with the linear extrapolation [11].

The third-order edge-based discretization (III.1) eliminates second-order errors in a special manner, relyingon the combination of the edge-based quadrature and the linear flux extrapolation. As discussed in Refs.[20,22],third-order accuracy is achieved by ensuring that the second-order truncation error contains only derivatives ofthe target equation. For example, the truncation error Tj for the inviscid terms on a regular grid composed ofright isosceles triangles can be obtained by substituting a smooth solution into the residual as

Tj =h2

12(∂xx + ∂xy + ∂yy) divF

i +O(h3), (III.2)

where h denotes the length of the leg of the right isosceles triangle. The second-order error term vanisheswhen the solution satisfies divFi = 0. This mechanism, often called the residual property and shared by othermethods such as the residual-compact method [29] and the residual-distribution method [30], implies that acareful discretization is necessary if other (not inviscid) terms are present. If source terms are present in thetarget equation (i.e., S = 0), then a special quadrature formula is required for the source term to preserve third-order accuracy. The source term S must be discretized in such a way that it yields the following second-ordertruncation error:

Tj =h2

12(∂xx + ∂xy + ∂yy)

(divFi − S

)+O(h3), (III.3)

so that the second-order error vanishes for exact solutions satisfying divFi − S = 0. The same treatment isapplied to the physical time derivatives, which can be incorporated as a source term for unsteady computations[12]. In this study, we focus on steady problems, and therefore relevant source terms are those generated bymanufactured solutions used for accuracy verification.

Two techniques are available to guarantee the property (III.3). One is an extended Galerkin formula [12],and the other is a general divergence formulation of source terms [20]; here we employ the latter. In the caseof Navier-Stokes FUN3D-i3rd discretization, it is clear that the second-order truncation error will not vanish ingeneral, except for the continuity equation (which has no viscous terms). Therefore, the overall accuracy willbe second-order in general [22]. Third-order accuracy is expected only in regions where the viscous terms arenegligibly small compared with the inviscid terms, i.e., divFi −S ≈ 0, for example, away from boundary layers.

III.C. Numerical Flux

The numerical flux is constructed as a sum of an inviscid flux Φijk and a viscous flux Φv

jk:

Φjk = Φijk +Φv

jk. (III.4)

For all methods, the inviscid flux is computed by the Roe flux:

Φijk =

1

2

[(Fi

n)R + (Fin)L

]− 1

2

∣∣Ain

∣∣ (UR −UL) , (III.5)

where the subscripts L and R indicate values at the left and right sides of the edge midpoint, and Fin and Ai

n

are the inviscid flux projected along the directed area vector and its Jacobian. The absolute Jacobian∣∣Ai

n

∣∣ isevaluated by the Roe-averages [41]. The viscous flux is given by the P1 continuous Galerkin discretization inFUN3D and FUN3D-i3rd. In the HNS method, the viscous flux is the upwind flux:

Φvjk =

1

2[(Fv

n)R + (Fvn)L]−

1

2P−1 (|PAv

n|+ |PAavn |) (UR −UL) , (III.6)

where Fvn and Av

n are the hyperbolic viscous flux projected along the directed area vector and its Jacobian,and |PAav

n | is the artificial hyperbolic dissipation required to avoid accuracy deterioration in the HNS velocity-gradient variables (II.2) [24, 26]. For the viscous flux, the dissipation matrix is evaluated by the arithmeticaverages. See Ref.[26] for more details.

5 of 41

American Institute of Aeronautics and Astronautics

Dow

nloa

ded

by H

iroa

ki N

ishi

kaw

a on

Jun

e 10

, 201

6 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.2

016-

3969

III.D. Solution Reconstruction and Flux Extrapolation for FUN3D and FUN3D-i3rd

The Galerkin viscous discretization does not require solution reconstruction at a face, and therefore thediscussion below is relevant to the inviscid fluxes only. The solutions are linearly reconstructed at the midpointof each edge. The reconstruction is performed for the primitive variables:

W = [ρ, u, v, w, p]. (III.7)

The left and right states WL and WR are reconstructed from the two end nodes of the edge [j, k]:

WL = Wj +1

2∇Wj ·∆ljk, WR = Wk − 1

2∇Wk ·∆ljk, (III.8)

where ∆ljk = (xk − xj , yk − yj , zk − zj), ∇Wj and ∇Wk are LSQ gradients of W computed at the nodes jand k, respectively. Linear and quadratic LSQ fits are used for FUN3D and FUN3D-i3rd schemes, respectively.The left and right conservative variables, UL and UR, are then obtained algebraically from WL and WR,respectively. A quadratic LSQ fit has been implemented in FUN3D by using a two-step implementation asdescribed in Ref.[22]. Unlike the linear fit, which is defined on a compact stencil (see Figure 3(a)), the quadraticfit has a larger stencil involving the neighbors of the neighbors (see Figure 3(b)). Nevertheless, in the two-stepimplementation, each step involves only the neighbors and therefore it is simple to implement in a parallel code.At many nodes in a tetrahedral grid, the number of neighbors is sufficient to form a valid quadratic fit. However,adding neighbors of the neighbors is necessary for robust iterative convergence.

In the FUN3D scheme, the left and right inviscid fluxes are evaluated from the reconstructed solutions:

(Fin)L = Fi

n (WL) , (Fin)R = Fi

n (WR) . (III.9)

In the FUN3D-i3rd scheme, the left and right inviscid fluxes are linearly extrapolated to the midpoint [8, 9]:

(Fin)L = (Fi

n)j +1

2

(∂Fi

n

∂W

)j

∇Wj ·∆ljk, (III.10)

(Fin)R = (Fi

n)k − 1

2

(∂Fi

n

∂W

)k

∇Wk ·∆ljk. (III.11)

It is emphasized that the flux extrapolation must be linear to achieve third-order accuracy; a quadratic extrap-olation leads to second-order accuracy [11,24].

III.E. Solution Reconstruction and Flux Extrapolation for HNS20-II and Improved HNS20-I

In the HNS schemes, the solution reconstruction is conducted for both the inviscid and viscous fluxes. Thelinear reconstruction (III.8) is performed for the following variables:

W = [ρ, u, v, w, p,g,q, r]. (III.12)

The left and right inviscid fluxes are linearly extrapolated as in Equations (III.10) and (III.11). The left andright viscous fluxes are evaluated from reconstructed solutions:

(Fvn)L = Fv

n (WL) , (Fvn)R = Fv

n (WR) . (III.13)

The gradient ∇W is computed at each node by the linear LSQ fit, and the resulting scheme is called Scheme-I[21] or HNS20-I to be specific to the system considered here. In the HNS method, a more economical constructionis possible because the gradient of the primitive variables [ρ, u, v, w, p] can be directly obtained from the gradientvariables:

∇ρj =rjνρ

, ∇vj =gj

(µv)j, ∇pj =

1

γ(ρj∇Tj + Tj∇ρj) = − (γ − 1)ρjqj

(µh)j+

Tjrjγνρ

. (III.14)

In this case, we do not need to compute LSQ gradients for the primitive variables [ρ, u, v, w, p]. This economicalversion is called Scheme-II [21] or HNS20-II. In the second-order HNS schemes, HNS20-I and HNS20-II, thegradients of the primitive-variable are obtained by the gradient variables with second-order accuracy. Therefore,

6 of 41

American Institute of Aeronautics and Astronautics

Dow

nloa

ded

by H

iroa

ki N

ishi

kaw

a on

Jun

e 10

, 201

6 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.2

016-

3969

the HNS20-II scheme achieves third-order accuracy in the inviscid terms (without using the quadratic fit). Thisis not possible with HNS20-I because the nodal gradients used in the reconstruction are computed by linearLSQ gradients which are only first-order accurate. However, it can be made possible by a simple modificationas we discuss below.

During the development of HNS schemes within FUN3D, we encountered a robustness issue with the HNS20-II scheme for high-Reynolds-number flows involving boundary layers. As a robust alternative to HNS20-II, wedeveloped an improved version of HNS20-I that can also achieve third-order accuracy in the inviscid terms.Instead of using the linear LSQ gradients for the primitive variables, we propose to construct quadratic LSQgradients by adding a curvature term to the linear LSQ formulation. Consider a quadratic polynomial of thedensity around a node j,

ρk = ρj +∇ρj ·∆ljk +1

2∆ltjkH

ρj∆ljk, (III.15)

where the superscript t indicates the transpose, k denotes a neighbor of the node j, and Hρj is the Hessian of

the density at j. The Hessian is the second-derivative matrix of ρ, and available in HNS20-I as the linear LSQgradient of r:

Hρj = ∇ (∇ρ)j = ∇

(r

νρ

)j

. (III.16)

Therefore, the only unknowns in Equation (III.15) are the gradients of ρ:

(xk − xj)(∂xρ)j + (yk − yj)(∂yρ)j + (zk − zj)(∂zρ)j = (ρk − ρj)−1

2∆ltjkH

ρj∆ljk. (III.17)

It, therefore, results in a linear LSQ problem over a compact stencil (see Figure 3(a)), and can be solved, in theLSQ sense, for (∂xρ)j , (∂yρ)j , and (∂zρ)j . In practice, the gradient is computed as

(∇ρ)j = [(∂xρ)j , (∂yρ)j , (∂zρ)j ] =∑

k∈{kj}

Cjk

(ρk − ρj −

1

2∆ltjkH

ρj∆ljk

), (III.18)

where Cjk is a vector of linear LSQ coefficients that are pre-computed and stored (see, e.g., Ref.[22]). It isemphasized that the resulting quadratic LSQ gradient ∇ρj depends only on the neighbors (unlike the quadraticLSQ gradient discussed in the previous section). Therefore, the residual stencil remains as compact as theFUN3D scheme, i.e., up to the neighbors of the neighbors. The same can be applied to the velocity and pressuregradients:

(∇u)j =∑

k∈{kj}

Cjk

(uk − uj −

1

2∆ltjkH

uj∆ljk

), (III.19)

(∇v)j =∑

k∈{kj}

Cjk

(vk − vj −

1

2∆ltjkH

vj∆ljk

), (III.20)

(∇w)j =∑

k∈{kj}

Cjk

(wk − wj −

1

2∆ltjkH

wj ∆ljk

), (III.21)

(∇p)j =∑

k∈{kj}

Cjk

(pk − pj −

1

2∆ltjkH

pj∆ljk

), (III.22)

with their Hessians directly evaluated at the node j as

Huj = ∇ (∇u)j =

(µv∇gu − gu⊗∇µv

µ2v

)j

, (III.23)

Hvj = ∇ (∇v)j =

(µv∇gv − gv⊗∇µv

µ2v

)j

, (III.24)

Hwj = ∇ (∇w)j =

(µv∇gw − gw⊗∇µv

µ2v

)j

, (III.25)

7 of 41

American Institute of Aeronautics and Astronautics

Dow

nloa

ded

by H

iroa

ki N

ishi

kaw

a on

Jun

e 10

, 201

6 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.2

016-

3969

Scheme

Solution W = (ρ,v, p) LSQ Gradient ∇WGradient variables

Hessian ∇∇W

General Inviscid General Inviscid (r,g,q) General Inviscid

FUN3D O(h2) O(h2) O(h) O(h) N/A O(1) O(1)

FUN3D-i3rd O(h2) O(h3) O(h) O(h2) N/A O(1) O(h)

HNS20-I O(h2) O(h2) O(h) O(h) O(h2) O(h) O(h)

HNS20-I(Q) O(h2) O(h3) O(h) O(h2) O(h2) O(h) O(h)

HNS20-II O(h2) O(h3) Not computed O(h2) O(h) O(h)

Table 2: Summary of expected orders of accuracy for the primitive variables W = (ρ,v, p) and their derivativeson unstructured grids; h denotes a typical mesh spacing. ’Inviscid’ means the inviscid limit, where the viscousterms are negligibly small. The Hessian is available as LSQ gradients of the gradient variables in HNS schemeswhile it can be computed by applying the LSQ method twice for W in FUN3D and FUN3D-i3rd.

Hpj = ∇ (∇p)j =

[−γ − 1

νρµh(q⊗r+ r⊗q)− ρ(γ − 1)

µh∇q− q⊗∇µh

µ2h

+T∇r

γνρ

]j

, (III.26)

where (∇gu)j , (∇gv)j , (∇gw)j , and (∇q)j are linear LSQ gradients of gu, gv, gw, and q, respectively (whichare already available in HNS20-I),⊗ denotes the dyadic product, and the gradient of the scaled viscosities areevaluated by

∇µv = ∇(4µ

3

)=

4

3

∂µ

∂T∇T = −4γ(γ − 1)q

3µh

∂µ

∂T, (III.27)

∇µh = ∇( γµ

Pr

)=

γ

Pr

∂µ

∂T∇T = −γ2(γ − 1)q

Prµh

∂µ

∂T. (III.28)

The resulting gradients will be second-order accurate in the inviscid limit where the primitive variables are ob-tained with third-order accuracy. This compact quadratic LSQ method is designated as c-quadratic (curvature-corrected quadratic fit). A particularly attractive feature of c-quadratic LSQ fits is its compact stencil equivalentto that of the linear LSQ fits. As a result, the resulting HNS scheme, called HNS20-I(Q), retains the same sten-cil extent as a second-order edge-based scheme (e.g., FUN3D) even with quadratic LSQ fits. See Table 1 forcomparison. HNS20-I(Q) has been found to be much more robust than HNS20-II for practical problems.

As summarized in Table 2, the HNS20 schemes produce second-order accurate gradients via the gradientvariables in general. As a consequence, the Hessian can be obtained with first-order accuracy by a linear LSQfit applied to the gradient variables. First-order accurate Hessians can be obtained in the inviscid limit byFUN3D-i3rd. It is important to note that on unstructured grids, quadratic LSQ gradients are second-orderaccurate only if the solution values are at least third-order accurate; the gradients are first-order accurate, inprinciple, if the solution values are second-order accurate. The entire quadratic-fit algorithm is not expectedto be exact for quadratic solutions unless the nodal solutions are computed by an algorithm that is exact forquadratic solutions, i.e., by a third-order scheme. From this point of view, second-order accuracy in the gradients(and first-order accuracy in the Hessian) is a special property of the second-order HNS schemes, which is rarelyobserved in other schemes.

IV. Boundary Treatment

IV.A. Third-Order Boundary Flux Quadrature

At a boundary node, the residual needs to be closed by boundary flux contributions unless boundary condi-tions are imposed strongly by directly specifying the solution values. Accuracy of the edge-based discretizationis then dictated by the boundary flux quadrature, and the form of the quadrature depends on the type ofelements adjacent to the boundary [11, 38, 42]. For first-order schemes, a point evaluation is sufficient, butfor second-order schemes, a special quadrature formula is required to guarantee linear exactness in the fluxintegration. A comprehensive list of second-order quadrature formulas for quadrilateral/triangular elements in2D and hexahedral/tetrahedral/prismatic/pyramidal elements in 3D can be found in Appendix E in Ref.[38].

8 of 41

American Institute of Aeronautics and Astronautics

Dow

nloa

ded

by H

iroa

ki N

ishi

kaw

a on

Jun

e 10

, 201

6 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.2

016-

3969

For third-order accuracy, these formulas should not be used [11]. To preserve third-order accuracy, a generalboundary-flux quadrature formula has been derived in Ref.[11]. Consider a dual face of a control volume arounda boundary node j as shown in Figure 4. The residual at the node j needs to be closed by the flux contributionacross the boundary dual face. The boundary flux contribution should be computed by the following quadratureformula: [

1

2Φjb(nB) +

1

4Φj2(nB) +

1

4Φj3(nB)

]|nB |, (IV.1)

where nB denotes the boundary element normal vector and nB = nB/|nB |. This is a general boundary-fluxquadrature formula on the dual boundary face [j, 2, 3] for tetrahedral grids: accuracy is preserved up to third-order in the boundary discretization [11]. Note that the coefficients 1/2 and 1/4 are misplaced in Ref.[11]; it iscorrected in the above.

The general boundary-flux quadrature formula guarantees that the flux integration is exact for quadraticfluxes on linear triangular boundary grids [11]. Therefore, high-order curved grids are not required for curvedgeometries. This is a very attractive feature for practical applications since third-order accuracy can be achievedwith existing tetrahedral grids used for second-order computations. However, high-order accurate surface normalvectors are required, in principle, for boundary conditions involving the surface normal direction [11]. Further-more, high-order integration is required to evaluate integrated quantities over a solid body of interest, e.g., thedrag coefficient, with high-order accuracy.

It is important to note that the fluxes Φj2 and Φj3 in Equation (IV.1) are computed at the midpointof the edge [j, 2] and [j, 3], respectively, with the linear reconstruction and flux extrapolation, similar to theinterior scheme (III.10) and (III.11), but with nB as the face normal vector. On the other hand, the flux Φjb isdetermined directly at the node j, incorporating physical boundary conditions, as discussed in the next section.

IV.B. Boundary Conditions

Physical boundary conditions are imposed weakly, except on viscous walls, through the numerical fluxevaluated at a boundary node, j, with the two states:

Φjb(nB) = Φjb(WL,WR), WL = Wj , WR = Wb, (IV.2)

where Wj is the vector of solution variables at the boundary node j, and Wb is the state specified by a boundarycondition. For example, at an outflow boundary, Wb is set by the solution variables at j with the pressurereplaced by the free stream pressure (the so-called back pressure condition). The boundary flux Φjb is thendetermined by the Roe flux, which recognizes the local characteristic directions and takes appropriate statesfor the flux computation. Alternatively, the flux Φjb may be directly specified by a boundary condition, whichis employed here for the inviscid slip-wall condition. Some representative boundary conditions are listed below(∞ subscript denotes free stream values):

1. Free stream:

Φjb(WL,WR) : Wb = [ρ∞, u∞, v∞, w∞, p∞,g∞,q∞, r∞]. (IV.3)

2. Subsonic outflow (Back pressure):

Φjb(WL,WR) : Wb = [ρj , uj , vj , wj , p∞,gj ,qj , rj ]. (IV.4)

3. Slip condition at inviscid wall (Hj is the specific total enthalpy at the node j):

Φjb = [ρj unB, ρj unB

vj + pjnB, ρj unBHj ]

t, unB= [vj − (vj · n∗)nB ] · nB . (IV.5)

4. No-slip condition at viscous adiabatic wall:

Φjb(WL,WR) : Wb = [ρj , 0, 0, 0, pj , gj , qj , rj ], gj = (gj · n∗)n∗, qj = qj − qj · n∗. (IV.6)

9 of 41

American Institute of Aeronautics and Astronautics

Dow

nloa

ded

by H

iroa

ki N

ishi

kaw

a on

Jun

e 10

, 201

6 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.2

016-

3969

In the above, the normal vector n∗, in the direction of which boundary conditions are imposed, needs to bedefined. Three different choices are considered for n∗:

n∗ =

nB Element normal ,

nexactj Exact normal at node j,

nlumpedj Lumped normal at node j.

(IV.7)

For third-order accuracy, the normal vector n∗ needs to be sufficiently accurate, i.e., exact for a quadraticgeometry [11]. However, in the current implementation, we employ the normal vector already available inFUN3D, which is exact only for linear geometries in general, and investigate the effect of the low-order accuratesurface normals. Exact normals can be determined analytically for geometries considered in this paper. Thelumped normal vector nlumped

j is a default normal at node available in FUN3D, which is defined as the unit vectorof the sum of the element-normals over the triangular boundary elements sharing the node. Implementation ofmore accurate surface normal vectors is left as future work.

At a node on a viscous wall, all schemes employ the strong condition:

uj = 0, vj = 0, wj = 0, (ρE)j =ρjTaw

γ(γ − 1), Taw = 1 +

γ − 1

2

√PrM2

∞, (IV.8)

where E is the specific total energy, Taw is an adiabatic wall temperature, and M∞ is the free stream Machnumber. In actual implementation, the residuals for the momentum and energy equations are first computedwith the weak boundary condition (IV.6), and then replaced by these algebraic equations (IV.8).

V. Numerical Results

V.A. Inviscid Results: FUN3D and FUN3D-i3rd

In this section, we consider the FUN3D and FUN3D-i3rd schemes to study the effect of the third-orderinviscid scheme and verify the boundary procedures. Both schemes are implemented in the FUN3D code, andall computations are performed with the FUN3D code. The implicit solver is taken to be converged when theroot-mean-square (RMS) norm of the residuals reaches 10−14. For all problems, we take γ = 1.4, Pr = 0.72,and T∞ = 300[K].

V.A.1. Accuracy verification

Accuracy of the third-order edge-based inviscid scheme has been verified by the method of manufacturedsolutions. To examine effects of the second-order Galerkin viscous scheme on third-order inviscid accuracy,we perform the verification with the Navier-Stokes equations for a range of Reynolds numbers. A smoothsine function is set as the exact solution by introducing source terms into the Navier-Stokes equations, withM∞ = 0.3. The source terms are discretized as described in Section III.B for preserving third-order accuracy.The discretization error is measured over a series of irregular tetrahedral grids with 83, 163, 323, and 643 nodesin a cubic domain with a strong implementation of the Dirichlet condition. A representative mesh spacing hV

is estimated as the L1 norm of the cubic root of the control volume. As shown in Figures 5 and 6 for thecoarsest grid, the grids are irregular with randomly perturbed interior nodes. Computations have been madefor a range of Reynolds numbers. Discretization error convergence results are shown for the primitive variables,(ρ, u, v, w, p) in Figures 7-11. Figures 7-9 show that second-order accuracy is observed in FUN3D-i3rd solutionsin low Reynolds number cases, and the errors are smaller than the FUN3D solutions, in particular, in the densityand the pressure. The improved FUN3D-i3rd accuracy could be attributed to the fact that the discretization ofthe continuity equation is third-order for all Reynolds numbers. For Re∞ = 100 and 1000, as shown in Figures10 and 11, where the inviscid terms dominate, third-order accuracy is observed for all the variables as expected.

V.A.2. Inviscid flow over a hemisphere cylinder

To verify third-order accuracy for a realistic configuration, an inviscid flow over a hemisphere cylinder isconsidered with M∞ = 0.3 and zero angle of attack. For this problem, the governing equations are the Eulerequations. Free stream boundary condition is applied at inflow, and the back-pressure condition is applied atoutflow. On the hemisphere-cylinder surface, the slip condition is imposed weakly as described in Section IV.B.

10 of 41

American Institute of Aeronautics and Astronautics

Dow

nloa

ded

by H

iroa

ki N

ishi

kaw

a on

Jun

e 10

, 201

6 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.2

016-

3969

For this problem, no exact solutions exist, and therefore we use spurious entropy generation as a measure ofthe error. Five tetrahedral grids with 110404, 262535, 535829, 927958, 1607469 nodes are used for accuracyverification. See Figure 12 for the coarsest grid. Consistent refinement check has been performed for thesegrids: a grid spacing estimated from the number of nodes, hN = N−1/3, where N is the number of nodes in thegrid, and a spacing hV computed as the L1 norm of the cubic root of the control volume have been confirmedto be reduced at the same rate in the grid refinement. Entropy error convergence results are shown in Figure13, where the L∞ norm of the entropy has been computed based on all the nodes in the domain. The FUN3Dscheme gives second-order convergence in the entropy as expected. The FUN3D-i3rd scheme gives third-orderconvergence with any of the normal vectors (IV.7) as long as the general boundary flux quadrature (IV.1) is used.The results indicate that all the surface normal vectors are accurate enough to produce third-order results. Toshow the importance of the boundary flux quadrature (IV.1), results obtained with the second-order boundaryflux quadrature [11, 38, 42] are also shown and indicated by FUN3D-i3rd (nB)-Q1. Clearly, only second-orderconvergence is observed although the entropy errors are smaller than those obtained by the FUN3D scheme.The maximum entropy error was found to occur at the stagnation point in the case of FUN3D, while it occurredat a node located in the junction of the hemisphere and the cylinder in the case of FUN3D-i3rd.

V.B. Viscous Results

In this section, we consider all schemes, FUN3D, FUN3D-i3rd, HNS20-I, HNS20-I(Q), and HNS20-II, forviscous flow problems. All schemes are implemented in the FUN3D code, and all computations are performedwith the FUN3D code. The implicit solver is taken to be converged when the RMS norm of residuals reaches10−12 unless otherwise stated. Again, we take γ = 1.4, Pr = 0.72, and T∞ = 300[K] for all problems considered.

V.B.1. Accuracy verification

Accuracy of the HNS schemes has been verified by the method of manufactured solutions as described inSection V.A.1, and compared with accuracy of the FUN3D and FUN3D-i3rd schemes. The grids are similarto those used in the previous case with 153, 203, 253, 303, 353, and 403 nodes randomly perturbed in theinterior as well as on the boundary planes. See Figures 14 and 15. Two different Reynolds numbers, Re∞ = 1and Re∞ = 104, are considered. Results are shown in Figure 16 for the pressure as a representative of theprimitive variables. Figure 16(a) shows that all solutions converge with second order for Re∞ = 1. FUN3D andHNS20-I errors are comparable and the largest among the five schemes. FUN3D-i3rd and HNS20-I(Q) errorsare comparable and much smaller than FUN3D and HNS20-I errors. The lowest error level is achieved by theHNS20-II scheme. Figure 16(b) shows the results for Re∞ = 104. Here, as expected, FUN3D-3rd, HNS20-I(Q), and HNS20-II solutions yield third-order accuracy in the pressure whereas FUN3D and HNS20-I solutionsremain second-order accurate. Again, the HNS20-II solution achieves the lowest error level, the HNS20-I(Q)solution comes next, and then the FUN3D-i3rd solution.

Gradient accuracy is shown in Figure 17 for ∂xp. Gradients are computed by linear LSQ fits in the FUN3Dscheme and by quadratic LSQ fits in the FUN3D-i3rd scheme. The gradient variables are used in the HNSschemes. Figure 17(a) shows the case Re∞ = 1. As expected, the second-order asymptotic convergence isobserved for the HNS schemes. The quadratic LSQ gradients in FUN3D-i3rd are more accurate than the linearLSQ gradients in FUN3D solutions and even than HNS20-I gradients, but the convergence rate deteriorates tofirst order on fine grids. Figure 17(b) shows results for Re∞ = 104, which confirm that the gradients are first-order accurate for the FUN3D scheme, and second-order accurate for the FUN3D-i3rd, HNS20-I, HNS20-I(Q)and HNS20-II schemes.

Accuracy of ∂xxp is shown in Figure 18. Second derivatives are computed by applying twice the linearLSQ fit in the FUN3D scheme and the quadratic LSQ fit in the FUN3D-i3rd scheme. LSQ gradients of thegradient variables are used in the HNS schemes. Figure 18(a) shows the case Re∞ = 1. As expected, thefirst-order asymptotic convergence is observed for the HNS schemes, and an inconsistent behavior is observed inthe FUN3D and FUN3D-i3rd schemes. Nevertheless, the FUN3D-i3rd scheme yields significantly more accurateHessian than the FUN3D and HNS20-I schemes. In the higher Reynolds number case, the Hessians computedby the FUN3D-i3rd and HNS schemes are first-order accurate as expected. See Figure 18(b). An interestingobservation here is that the gradient error for Re∞ = 104 is higher in the FUN3D-i3rd scheme than the HNSschemes, but the Hessian error is much smaller in the FUN3D-i3rd scheme than the HNS schemes.

11 of 41

American Institute of Aeronautics and Astronautics

Dow

nloa

ded

by H

iroa

ki N

ishi

kaw

a on

Jun

e 10

, 201

6 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.2

016-

3969

Grid Nodes FUN3D FUN3D-i3rd HNS20-I HNS20-I(Q)

Grid1 5,668 1.3865 1.5663 1.4229 1.3841

Grid2 50,228 1.1444 1.1895 1.1521 1.1611

Grid3 466,830 1.0844 1.0926 1.0943 1.0976

Grid4 1,656,714 1.0772 1.0808 1.0890 1.0905

Table 3: Drag coefficient convergence for sphere computations. The residuals converged to the tolerance within3,000 iterations in all cases except FUN3D and FUN3D-i3rd for Grid4, which did not reach the tolerance in30,000 iterations and thus stopped at 30,000 iterations.

Grid NodesFUN3D FUN3D-i3rd HNS20-I HNS20-I(Q)

(CD)p (CD)vis (CD)p (CD)vis (CD)p (CD)vis (CD)p (CD)vis

Grid1 5,668 0.77036 0.61616 0.92860 0.63770 0.80500 0.61787 0.80496 0.57910

Grid2 50,228 0.56020 0.58417 0.63400 0.55554 0.53879 0.61329 0.57225 0.58890

Grid3 466,830 0.51854 0.56590 0.53738 0.55522 0.50622 0.58809 0.51394 0.58362

Grid4 1,656,714 0.51303 0.56413 0.52149 0.55927 0.50651 0.58250 0.51001 0.58052

Table 4: Drag decomposition for sphere computations.

V.B.2. Gradients on skewed grid

To demonstrate the quality of the gradients predicted by the HNS schemes, we performed computationson a highly skewed grid in a rectangular domain (x, y, z) ∈ [0, 1] × [0, 1] × [0, 0.005] shown in Figure 19. Theparameters are taken as M∞ = 0.5 and Re∞ = 25. The grid has 33×33×33 nodes randomly perturbed in y andz directions Figure 20 shows the section of the grid at x = 0.5. We focus on the gradient component ∂zu, whichis relevant to the viscous stress component in a boundary layer. Results are summarized in Figure 21. Thecontour plot of the exact values of ∂zu is given in Figure 21(a). All contours have been plotted within the rangeof the exact contours. The gradients are computed by linear LSQ fits in FUN3D, quadratic LSQ fits in FUN3D-i3rd, and by the gradient variables in the HNS schemes. These contour plots show that the HNS schemes canproduce a smooth and accurate derivative on a highly-skewed tetrahedral grid, and the LSQ gradients are veryinaccurate and far beyond the exact range. The quadratic LSQ gradient is more accurate than the linear LSQgradient, but it still shows irregular contours. The ranges of the HNS derivatives match very well the range ofthe exact derivative. The HNS20-I(Q) scheme produces smoother contours than the HNS20-I scheme.

These results indicate that the HNS schemes will be useful for turbulent-flow solutions that are highlysensitive to accuracy of gradients used for the source terms in turbulence models. For such applications, schemesemploying LSQ methods for gradient evaluations may encounter difficulties on adapted viscous tetrahedral grids.

V.B.3. Low-Reynolds-number flow over a sphere (Re∞ = 101)

To study effects of third-order inviscid accuracy for low-Reynolds number flows over curved geometries, weconsider a laminar flow over a sphere with M∞ = 0.15 and Re∞ = 101. The Reynolds number is definedbased on the diameter of the sphere. An experimental value of the drag coefficient is given by CD = 1.08 [43].This is similar to the case considered for the HNS20-II scheme in Ref.[26], but the grids are purely tetrahedraland severely distorted with random perturbation applied to interior nodes (see Figure 22). The sphere surfaceis systematically triangulated with no singularities (see Figure 23), and a similar triangulation is used on theouter boundary, which is also a spherical surface and located at the distance of 100 times the diameter of thesphere. The viscous wall condition and the free stream condition are applied to the inner and outer boundaries,respectively. Four levels of tetrahedral grids have been generated with randomly perturbed nodes: 5668, 50228,466830, and 1656714 nodes. The coarsest one is shown in Figures 22 and 23.

The HNS20-II scheme has been found to suffer from convergence difficulties, and therefore no results areavailable (It runs fine for grids without nodal perturbations). Results obtained by other schemes are summarized

12 of 41

American Institute of Aeronautics and Astronautics

Dow

nloa

ded

by H

iroa

ki N

ishi

kaw

a on

Jun

e 10

, 201

6 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.2

016-

3969

Grid Nodes FUN3D FUN3D-i3rd HNS20-I HNS20-I(Q)

Grid1 9,720 0.26867 0.25725 0.26700 0.25286

Grid2 64,400 0.25856 0.25702 0.25496 0.25250

Grid3 462,240 0.25427 0.25440 0.25217 0.25186

Grid4 1,500,720 0.25234 0.25252 0.25174 0.25167

Table 5: Drag coefficient convergence for Joukowsky airfoil computations. The residuals converged to thetolerance within 3,600 iterations in all cases.

Grid NodesFUN3D FUN3D-i3rd HNS20-I HNS20-I(Q)

(CD)p (CD)vis (CD)p (CD)vis (CD)p (CD)vis (CD)p (CD)vis

Grid1 9,720 0.0984 0.17025 0.0920 0.16528 0.0988 0.16814 0.0904 0.16246

Grid2 64,400 0.0907 0.16784 0.0890 0.16800 0.0901 0.16482 0.0884 0.16413

Grid3 462,240 0.0890 0.16525 0.0889 0.16548 0.0882 0.16397 0.0880 0.16390

Grid4 1,500,720 0.0887 0.16363 0.0889 0.16358 0.0880 0.16370 0.0880 0.16367

Table 6: Drag decomposition for Joukowsky airfoil computations.

in Figures 24-29, and Tables 3 and 4. Surface pressure contours on the finest grid are shown in Figure 24. Nomajor differences can be seen on the surface pressure, but the contours in the interior domain reveal a strikingdifference between second- and third-order inviscid schemes. As shown in Figure 25, the second-order schemes,FUN3D and HNS20-I, yield noisy contours in the domain, but the third-order inviscid schemes, FUN3D-i3rdand HNS20-I(Q) produce very smooth pressure distributions. This feature has been observed previously forHNS20-II on irregular triangular grids in Ref.[24] and on tetrahedral-prismatic mixed grids in Ref.[26].

To examine gradient accuracy, we compare the vorticity contours in Figures 26 and 27. Surface vorticitycontour plots in Figure 26 show that HNS20 schemes yield, as expected, very smooth distributions whereasFUN3D and FUN3D-i3rd yield noisy vorticity contours. The same can be seen in the interior domain as shownin Figure 27. It is observed that the HNS20-I(Q) scheme gives slightly smoother contours than the HNS20-Ischeme; the inviscid approximation is third-order in the former and second-order in the latter. Observe alsothat quadratic LSQ gradients in the FUN3D-i3rd scheme do not seem to bring significant improvement over thelinear LSQ gradients in the FUN3D schemes for the vorticity.

Despite the striking differences in the pressure and vorticity contours, no major differences are observed inthe drag coefficient for fine grids as shown in Table 3 and Figure 28. Similarly, no significant differences areobserved in the pressure and viscous drag components (see Table 4 and Figure 29). See also Figures 28(a) and28(b). It seems to indicate that the low-order force integration algorithm, which assumes linear variation inthe pressure and the viscous stresses (the default quadrature in FUN3D), has a large impact on accuracy ofintegrated quantities over a spherical surface, not benefiting from accurate viscous stresses produced by the HNSschemes on the surface. In the HNS schemes, it can be improved by using a higher-order quadrature, whichis possible because the first-order accurate gradients of the viscous stresses are available at boundary nodes.Combined with a high-order local surface reconstruction, it may bring improvements in the drag prediction overa spherical surface and other curved geometries.

V.B.4. Moderate-Reynolds-number flow over a Joukowsky airfoil (Re∞ = 1, 000)

A laminar flow over a Joukowsky airfoil at the angle of attack 2◦ is considered for a moderate Reynoldsnumber of 1000. The Reynolds number is defined based on the chord length of the airfoil. The flow conditionsare M∞ = 0.5 and Re∞ = 1, 000. The Joukowsky airfoil is defined by a set of parameters: ℓ = 0.25, ϵ = 0.3,and κ = 0.0 (see Ref.[31] for the definitions of the parameters). The outer boundary is a circle located at thedistance of 100 times the chord length. The viscous wall condition and the free stream condition are applied tothe airfoil surface and the outer boundary, respectively. On the two boundary planes located at y = 0 and y = 2,a periodic boundary condition is imposed. Note that the domain is 3D, and therefore the case is equivalent to a

13 of 41

American Institute of Aeronautics and Astronautics

Dow

nloa

ded

by H

iroa

ki N

ishi

kaw

a on

Jun

e 10

, 201

6 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.2

016-

3969

3D wing of infinite span. Four levels of tetrahedral grids have been generated with nodes randomly perturbedwithin (x, z)-plane: 9720, 64400, 462240, and 1500720 nodes; 81, 161, 321, 481 nodes on the airfoil. The coarsestgrid is shown in Figures 30 and 31. It is emphasized that the grid lines are not orthogonal to the viscous surfacedue to the random perturbation, modeling adaptive viscous grids. For this problem, the tolerance is set tobe 10−10. The quadratic LSQ matrix was found to be singular at some nodes in Grid3 and Grid4, and as atemporary fix, such quadratic LSQ fits were replaced by linear LSQ fits. This fix was necessary to obtain resultsfor the FUN3D-i3rd scheme. As in the sphere case, the HNS20-II scheme suffers from convergence difficulties,and therefore no results are available.

Results are summarized in Tables 5 and 6, and Figures 32-39. As in the sphere case, the third-order inviscidschemes, FUN3D-i3rd and HNS20-I(Q), yield much smoother pressure contours than second-order schemes,which can be seen in Figure 32 for Grid 1, and in Figure 35 for Grid 2. Also, superior gradient accuracy by theHNS schemes can be seen in the vorticity-magnitude contours as shown in Figure 33 for Grid 1, and in Figure 36for Grid 2. The vorticity magnitude |ω| = |curlu| was computed by LSQ gradients in the FUN3D and FUN3D-i3rd schemes, and by the gradient variables as |curlu| = |Egt| in the HNS schemes, where E is the third-rankalternating tensor of the Eddington epsilon [31]. Hessian contours are also shown for ∂zzu in Figures 34 and 37.Observe that the FUN3D-i3rd and HNS schemes produce significantly more accurate Hessian contours than theFUN3D scheme. Also observed is that the FUN3D-i3rd scheme gives more accurate and smooth Hessian thanthe HNS schemes, which is consistent with the observation made in the accuracy verification study in SectionV.B.1.

For this problem, the HNS schemes provide a significant improvement in the drag prediction. Table 5 andFigure 38 show that the drag coefficient converges very rapidly especially with the HNS20-I(Q) scheme. TheFUN3D-i3rd and FUN3D schemes give similar results for fine grids because the quadratic LSQ fits encountered aproblem at many nodes and were replaced by linear LSQ fits. A more sensible comparison based on the numberof degrees of freedom (DoFs) is shown in Figure 38(b). Here, the mesh spacing hDoF is defined as hDoF = N−1/3

for the FUN3D and FUN3D-i3rd schemes, where N is the total number of nodes; and hDoF = (4N)−1/3

for theHNS schemes, which carry four times as many variables as the FUN3D and FUN3D-i3rd schemes. Figure 38(b)shows that the HNS20-I(Q) scheme gives more accurate drag prediction than others even for the same numberof DoFs, i.e., for the same number of discrete solution values. In contrast to the sphere case, the viscous dragdominates for this type of slender geometry. This is confirmed by the pressure and viscous drag decompositionas shown in Table 6 and Figure 39. The results indicate that improvements for integrated quantities can stillbe expected without high-order quadrature.

V.B.5. High-Reynolds-number flow over a flat plate (Re∞ = 1, 000, 000)

A laminar flow over a flat plate is considered for a high Reynolds number of 1 million. The flow conditionsare M∞ = 0.5 and Re∞ = 106. The Reynolds number is defined based on the length (in the x-direction) ofthe flat plate. The domain is taken to be rectangular (x, y, z) ∈ [−2, 1]× [0, 0.5]× [0, 3], where the flat plate islocated at the bottom (z = 0) and (x, y) ∈ [0, 1] × [0, 0.5]. Seven tetrahedral grids are generated: 3384, 8064,65024, 238368, 554880, 1071840, and 1838400 nodes. Nodal perturbation is not applied to these grids, and thusthe grids are relatively regular. Figure 40 shows the coarsest grid. A parabolic stretching has been appliedacross the boundary layer in order to resolve the boundary layer equally along the flat plate (see Figure 40(b)).The cell aspect ratio is of O(103) in the boundary layer region over the flat plate. A reference value of the dragcoefficient is 0.001328 based on a theory for incompressible viscous flows [44]. A symmetry boundary conditionis used at boundary planes at y = 0 and y = 0.5, and also at z = 0 ahead of the flat plate. Free stream conditionis used at the inflow boundary (x = −2), and the back pressure condition is used at the top (z = 3) and outflow(x = 1) boundaries. Again, no results are available for the HNS20-II scheme due to convergence difficulties.

Despite a regular structure of the grids, third-order schemes encountered issues. Quadratic LSQ fits in theFUN3D-i3rd scheme failed at some nodes and were reverted to linear LSQ fits as described in the previous section.In the HNS20-I(Q) scheme, a singularity at the leading edge was found to cause difficulties in convergence. Again,a temporary fix was introduced, which turns off the Hessian correction in the c-quadratic fit in a small regionwithin the distance of 10 times the boundary layer thickness 5.5/

√Re∞ from the leading edge. Then, the

HNS20-I(Q) scheme converged for all cases; a smaller CFL number of 100 had to be used for Grid4 and Grid 7.The FUN3D and HNS20-I schemes are quite robust, and results were obtained on all grids with no difficulties.

Comparing the HNS schemes and the FUN3D scheme in Table 7 and Figure 41, one can see that theHNS schemes, especially HNS20-I(Q), give a superior drag convergence. It indicates that the surface integrationaccuracy has no large impact on integrated quantities on such a non-curved geometry. Hence, improved gradient

14 of 41

American Institute of Aeronautics and Astronautics

Dow

nloa

ded

by H

iroa

ki N

ishi

kaw

a on

Jun

e 10

, 201

6 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.2

016-

3969

Grid Nodes FUN3D FUN3D-i3rd HNS20-I HNS20-I(Q)

Grid1 3,384 0.0011894 0.0011986 0.0012667 0.0013249

Grid2 8,064 0.0012292 0.0012315 0.0012923 0.0013294

Grid3 65,024 0.0012740 0.0012737 0.0013169 0.0013234

Grid4 238,368 0.0012890 0.0012889 0.0013222 0.0013249

Grid5 554,880 0.0012968 0.0012970 0.0013243 0.0013257

Grid6 1,071,840 0.0013020 0.0013022 0.0013255 0.0013261

Grid7 1,838,400 0.0013053 0.0013059 0.0013261 0.0013264

Table 7: Drag coefficient convergence for 3D flat plate computations. CFL = 100 for HNS20-I(Q) on Grid4 andGrid7. The residuals converged to the tolerance within 25,000 iterations in all cases.

accuracy by the HNS schemes is immediately observed in the drag coefficient. The FUN3D-i3rd scheme, onthe other hand, does not show improvements over the FUN3D scheme, apparently because quadratic LSQ fitswere replaced by linear LSQ fits in a region having a large impact on the drag coefficient. A more practicalcomparison is shown in Figure 41(b). It clearly indicates that the HNS schemes yield more accurate dragprediction for the same number of DoFs (i.e., for the same discrete problem size). It is noted that the grids arenot perturbed and thus quite regular, but the order of accuracy in the gradients at the wall is first-order in theFUN3D scheme and second-order in the HNS schemes.

VI. Concluding Remarks

The initial development of third-order edge-based schemes in NASA’s FUN3D code has been presented.Purely inviscid schemes have been compared for second- and third-order accuracy. Third-order accuracy hasbeen verified for the inviscid terms by the method of manufactured solutions and by convergence of the entropyerror for a smooth flow over a hemisphere cylinder. Hyperbolic Navier-Stokes (HNS) schemes have been verifiedand tested for laminar test cases. An improved version of an HNS scheme has been introduced and found to berobust and accurate for the laminar cases considered. It was found that pressure distributions were improvedby third-order inviscid schemes even for relatively low Reynolds number flows. The gradient accuracy was alsofound to be improved by the third-order inviscid scheme, and greater improvements were observed by the HNSschemes for practical cases. Grid convergence results for drag prediction in the airfoil and flat plate cases showthat HNS schemes can provide more accurate predictions for the same number of discrete unknowns (degreesof freedom).

The numerical results indicate that the HNS schemes will be useful especially on fully adaptive viscous grids,i.e., grid adaptation can be applied in the entire domain including boundary layers without degrading accuracyin both the solution and the gradients. Moreover, the HNS schemes provide a first-order accurate Hessian,which is expected to provide a more reliable guide for anisotropic grid adaptation [45]. Note that conventionalsecond-order schemes can provide no (i.e., zero-th order) accuracy for Hessian on unstructured grids. Thesuperior gradient prediction capability is expected to play a key role in various other areas: turbulence modeldiscretizations, which require accurate evaluations of the vorticity or the strain tensor in their source terms,a hypersonic heating prediction problem on tetrahedral grids [46], etc. Studies aiming at improvements toquadratic LSQ gradients on viscous grids, high-order force integration on curved geometries, a theoreticalinvestigation for high-Reynolds-number issues for HNS schemes, and extensions to unsteady flows are currentlyunderway and will be reported in subsequent papers.

Acknowledgments

This work has been funded by the U.S. Army Research Office under the contract/grant number W911NF-12-1-0154 with Dr. Matthew Munson as the program manager, and by NASA under Contract No. NNL09AA00Awith Dr. Veer N. Vatsa as the technical monitor. The second author gratefully acknowledges support fromSoftware CRADLE. The authors would like to thank Michael A. Park (NASA Langley Research Center) andBoris Diskin (National Institute of Aerospace) for their valuable comments.

15 of 41

American Institute of Aeronautics and Astronautics

Dow

nloa

ded

by H

iroa

ki N

ishi

kaw

a on

Jun

e 10

, 201

6 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.2

016-

3969

References

1Biedron, R. T., Carlson, J.-R., Derlaga, J. M., Gnoffo, P. A., Hammond, D. P., Jones, W. T., Kleb, B., Lee-Rausch, E. M.,Nielsen, E. J., Park, M. A., Rumsey, C. L., Thomas, J. L., and Wood, W. A., “FUN3D Manual: 12.9,” NASA-TM–2016–219012 ,February 2016.

2Nakashima, Y., Watanabe, N., and Nishikawa, H., “Development of an Effective Implicit Solver for General-Purpose Un-structured CFD Software,” The 28th Computational Fluid Dynamics Symposium, C08-1, Tokyo, Japan, 2014.

3Kroll, N., Abu-Zurayk, M., Dimitrov, D., Franz, T., Fuhrer, T., Gerhold, T., Gortz, S., Heinrich, R., Ilic, C., Jepsen, J.,Jagerskupper, J., Kruse, M., Krumbein, A., Langer, S., Liu, D., Liepelt, R., Reimer, L., Ritter, M., Schwoppe, A., Scherer, J.,Spiering, F., Thormann, R., Togiti, V., Vollmer, D., and Wendisch, J.-H., “DLR Project Digital-X: Towards Virtual Aircraft Designand Flight Testing Based on High-Fidelity Methods,” CEAS Aeronautical Journal , Vol. 7, No. 1, March 2016, pp. 3–27.

4Mavriplis, D. J. and Long, M., “NSU3D Results for the Fourth AIAA Drag Prediction Workshop,” AIAA Paper 2010-4363,2010.

5Luo, H., Baum, J. D., and Lohner, R., “High-Reynolds Number Viscous Flow Computations using an Unstructured-GridMethod,” Proc. of 42nd AIAA Aerospace Sciences Meeting, AIAA Paper 2004-1103, Reno, Nevada, 2004.

6Kozubskaya, T., Abalakin, I., Dervieux, A., and Ouvrard, H., “Accuracy Improvement for Finite-Volume Vertex-CenteredSchemes Solving Aeroacoustics Problems on Unstructured Meshes,” Proc. of 16th AIAA/CEAS Aeroacoustics Conference, AIAAPaper 2010-3933, 2010.

7Haselbacher, A. and Blazek, J., “Accurate and Efficient Discretization of Navier-Stokes Equations on Mixed Grids,” AIAAJ., Vol. 38, No. 11, 2000, pp. 2094–2102.

8Katz, A. and Sankaran, V., “Mesh Quality Effects on the Accuracy of CFD Solutions on Unstructured Meshes,” J. Comput.Phys., Vol. 230, 2011, pp. 7670–7686.

9Katz, A. and Sankaran, V., “An Efficient Correction Method to Obtain a Formally Third-Order Accurate Flow Solver forNode-Centered Unstructured Grids,” J. Sci. Comput., Vol. 51, 2012, pp. 375–393.

10Diskin, B. and Thomas, J. L., “Effects of Mesh Regularity on Accuracy of Finite-Volume Schemes,” Proc. of 50th AIAAAerospace Sciences Meeting, AIAA Paper 2012-0609, Nashville, Tennessee, 2012.

11Nishikawa, H., “Accuracy-Preserving Boundary Flux Quadrature for Finite-Volume Discretization on Unstructured Grids,”J. Comput. Phys., Vol. 281, 2015, pp. 518–555.

12Pincock, B. and Katz, A., “High-Order Flux Correction for Viscous Flows on Arbitrary Unstructured Grids,” J. Sci. Comput.,Vol. 61, 2014, pp. 454–476.

13Tong, O., Katz, A., Yanagita, Y., Casey, A., and Schaap, R., “High-Order Methods for Turbulent Flows on Three-DimensionalStrand Grids,” J. Sci. Comput., 2015, pp. 1–19.

14Katz, A. and Work, D., “High-Order Flux Correction/Finite Difference Schemes for Strand Grids,” J. Comput. Phys.,Vol. 282, 2015, pp. 360–380.

15Park , M. A. and Darmofal, D. L., “Parallel Anisotropic Tetrahedral Adaptation,” Proc. of 46th AIAA Aerospace SciencesMeeting and Exhibit , AIAA Paper 2008-917, Reno, Nevada, 2008.

16Yano, M. and Darmofal, D. L., “An Optimization-Based Framework for Anisotropic Simplex Mesh Adaptation,” Journal ofComputational Physics, Vol. 231, No. 22, Sept. 2012, pp. 7626–7649.

17Michal, T. and Krakos, J., “Anisotropic Mesh Adaptation Through Edge Primitive Operations,” Proc. of 50th AIAAAerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition, AIAA Paper 2012-0159, Nashville,Tennessee, 2012.

18Alauzet, F. and Loseille, A., “A Decade of Progress on Anisotropic Mesh Adaptation for Computational Fluid Dynamics,”Computer-Aided Design, March 2015, pp. 13–39.

19Park, M. A., Krakos, J. A., Michal, T., Loseille, A., and Alonso, J. J., “Unstructured Grid Adaptation: Status, PotentialImpacts, and Recommended Investments Toward CFD Vision 2030,” AIAA AVIATION 2016 Forum and Exposition, AmericanInstitute of Aeronautics and Astronautics, Reston, VA (manuscript in preparation).

20Nishikawa, H., “Divergence Formulation of Source Term,” J. Comput. Phys., Vol. 231, 2012, pp. 6393–6400.21Nishikawa, H., “First-, Second-, and Third-Order Finite-Volume Schemes for Diffusion,” J. Comput. Phys., Vol. 256, 2014,

pp. 791–805.22Nishikawa, H., “First, Second, and Third Order Finite-Volume Schemes for Advection-Diffusion,” J. Comput. Phys., Vol. 273,

2014, pp. 287–309.23Nishikawa, H., “First, Second, and Third Order Finite-Volume Schemes for Navier-Stokes Equations,” Proc. of 7th AIAA

Theoretical Fluid Mechanics Conference, AIAA Aviation and Aeronautics Forum and Exposition 2014 , AIAA Paper 2014-2091,Atlanta, GA, 2014.

24Nishikawa, H., “Alternative Formulations for First-, Second-, and Third-Order Hyperbolic Navier-Stokes Schemes,” Proc. of22nd AIAA Computational Fluid Dynamics Conference, AIAA Paper 2015-2451, Dallas, TX, 2015.

25Nishikawa, H., “New-Generation Hyperbolic Navier-Stokes Schemes: O(1/h) Speed-Up and Accurate Viscous/Heat Fluxes,”Proc. of 20th AIAA Computational Fluid Dynamics Conference, AIAA Paper 2011-3043, Honolulu, Hawaii, 2011.

26Nakashima, Y., Watanabe, N., and Nishikawa, H., “Hyperbolic Navier-Stokes Solver for Three-Dimensional Flows,” 54thAIAA Aerospace Sciences Meeting, AIAA Paper 2016-1101, San Diego, CA, 2016.

27Abalakin, I., Bakhvalov, P., and Kozubskaya, T., “Edge-Based Reconstruction Schemes for Prediction of Near Field FlowRegion in Complex Aeroacoustics Problems,” Int. J. Aeroacoust., Vol. 13, 2014, pp. 207–233.

28Yang, H. Q. and Harris, R. E., “Vertex-Centered, High-Order Schemes for Turbulent Flows,” Proc. of 54th AIAA AerospaceSciences Meeting, AIAA Paper 2016-1098, San Diego, CA, 2016.

29Corre, C., Hanss, G., and Lerat, A., “A Residual-Based Compact Schemes for the Unsteady Compressible Navier-StokesEquations,” Comput. Fluids, Vol. 34, 2005, pp. 561–580.

30Nishikawa, H. and Roe, P. L., “On High-Order Fluctuation-Splitting Schemes for Navier-Stokes Equations,” ComputationalFluid Dynamics 2004 , edited by C. Groth and D. W. Zingg, Springer-Verlag, 2004, pp. 799–804.

16 of 41

American Institute of Aeronautics and Astronautics

Dow

nloa

ded

by H

iroa

ki N

ishi

kaw

a on

Jun

e 10

, 201

6 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.2

016-

3969

31Masatsuka, K., “I do like CFD, VOL.1, Second Edition,” http://www.cfdbooks.com, 2013.32Nakashima, Y., Private Communication, 2015.33Baty, H. and Nishikawa, H., “Hyperbolic Method for Magnetic Reconnection Process in Steady State Magnetohydrodynam-

ics,” Mon. Not. R. Astron. Soc., Vol. 459, 2016, pp. 624–637.34Nishikawa, H., “A First-Order System Approach for Diffusion Equation. II: Unification of Advection and Diffusion,” J.

Comput. Phys., Vol. 229, 2010, pp. 3989–4016.35Nishikawa, H., Diskin, B., Thomas, J. L., and Hammond, D. H., “Recent Advances in Agglomerated Multigrid,” 51st AIAA

Aerospace Sciences Meeting, AIAA Paper 2013-863, Texas, 2013.36Pandya, M. J., Diskin, B., Thomas, J., and Frink, N. T., “Assessment of Preconditioner for a USM3D Hierarchical Adaptive

Nonlinear Iteration Method (HANIM),” 54th AIAA Aerospace Sciences Meeting, AIAA Paper 2016-0860, San Diego, CA, 2016.37Diskin, B. and Thomas, J. L., “Accuracy Analysis for Mixed-Element Finite-Volume Discretization Schemes,” NIA Report

No. 2007-08 , 2007.38Nishikawa, H., “Beyond Interface Gradient: A General Principle for Constructing Diffusion Schemes,” Proc. of 40th AIAA

Fluid Dynamics Conference and Exhibit , AIAA Paper 2010-5093, Chicago, 2010.39Diskin, B. and Thomas, J. L., “Comparison of Node-Centered and Cell-Centered Unstructured Finite-Volume Discretizations:

Inviscid Fluxes,” AIAA J., Vol. 49, No. 4, 2011, pp. 836–854.40Ollivier Gooch, C., Nejat, A., and Michalak, K., “On Obtaining High-Order Finite-Volume Solutions to the Euler Equations

on Unstructured Meshes,” Proc. of 18th AIAA Computational Fluid Dynamics Conference, AIAA Paper 2007-4464, Miami, 2007.41Roe, P. L., “Approximate Riemann Solvers, Parameter Vectors, and Difference Schemes,” J. Comput. Phys., Vol. 43, 1981,

pp. 357–372.42Luo, H., Baum, J. D., and Lohner, R., “An Improved Finite Volume Scheme for Compressible Flows on Unstructured Grids,”

Proc. of 33rd Aerospace Sciences Meeting and Exhibit , AIAA Paper 95-0348, 1995.43Roos, F. W. and Willmarth, W. W., “Some Experimental Results on Sphere and Disk Drag,” AIAA J., Vol. 9, No. 2,

February 1971, pp. 285–291.44White, F. M., Viscous Fluid Flow , McGraw-Hill, 2nd ed., 1991.45Park, M. A., Loseille, A., Krakos, J. A., and Michal, T. R., “Comparing Anisotropic Output-Based Grid Adaptation Methods

by Decomposition,” Proc. of 22nd AIAA Computational Fluid Dynamics Conference, AIAA Paper 2015-2292, Dallas, TX, 2015.46Gnoffo, P. A. and White, J. A., “Computational Aerothermodynamic Simulation Issues on Unstructured Grids,” AIAA Paper

2004–2371, 2004.

j

k

c

m

cR

cL

(a) Dual face contributions from an adjacent tetrahedralelement to the edge [j, k].

j

k

(b) Total dual face at the edge [j, k].

Figure 1: Dual face contribution at the edge [j, k]. A numerical flux is evaluated at the midpoint of the edgem, indicated by an open circle. The centroid of the tetrahedral element is denoted by c, and the centroids ofthe two adjacent triangles are denoted by cL and cR.

17 of 41

American Institute of Aeronautics and Astronautics

Dow

nloa

ded

by H

iroa

ki N

ishi

kaw

a on

Jun

e 10

, 201

6 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.2

016-

3969

(a) Second-order finite-volume discretization (b) Second-order edge-based discretization

(c) Third-order finite-volume discretization (d) Third-order edge-based discretization

Figure 2: Illustration of quadrature points (open circles) required for second- and third-order accurate finite-volume and edge-based discretizations. Third-order edge-based discretization does not require any additionalquadrature points over the second-order discretization.

18 of 41

American Institute of Aeronautics and Astronautics

Dow

nloa

ded

by H

iroa

ki N

ishi

kaw

a on

Jun

e 10

, 201

6 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.2

016-

3969

(a) Stencil for linear and c-quadratic LSQ fit. (b) Stencil for quadratic LSQ fit.

Figure 3: Stencils for LSQ fits illustrated on a triangular grid. Gradients are computed at the node indicatedby a large red filled circle with the neighbors indicated by the black filled circles. Curvature-corrected quadraticLSQ fit (c-quadratic) developed for the HNS20-I(Q) scheme has a compact stencil equivalent to that of thelinear LSQ fit.

j, b

2

3

nB

|

|

|||

|||

Figure 4: A dual-face (shaded area) of a control volume around a boundary node j. The boundarydual-face normal vector nB is pointing outward from the interior domain.

19 of 41

American Institute of Aeronautics and Astronautics

Dow

nloa

ded

by H

iroa

ki N

ishi

kaw

a on

Jun

e 10

, 201

6 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.2

016-

3969

Figure 5: Irregular cube grid (163 nodes) for accuracyverification study in Section V.A.1.

Figure 6: Slices of the irregular cube grid (163 nodes)for accuracy verification study in Section V.A.1.

-2 -1.5 -1 -0.5 0

hV

-6

-5

-4

-3

-2

-1

L2error

ρ(FUN3D)u(FUN3D)v(FUN3D)w(FUN3D)p(FUN3D)ρ(FUN3D-i3rd)u(FUN3D-i3rd)v(FUN3D-i3rd)w(FUN3D-i3rd)p(FUN3D-i3rd)Slope 2Slope 3

Figure 7: Error convergence for Re∞ = 0.1.

-2 -1.5 -1 -0.5 0

hV

-6

-5

-4

-3

-2

-1

L2error

ρ(FUN3D)u(FUN3D)v(FUN3D)w(FUN3D)p(FUN3D)ρ(FUN3D-i3rd)u(FUN3D-i3rd)v(FUN3D-i3rd)w(FUN3D-i3rd)p(FUN3D-i3rd)Slope 2Slope 3

Figure 8: Error convergence for Re∞ = 1.0.

20 of 41

American Institute of Aeronautics and Astronautics

Dow

nloa

ded

by H

iroa

ki N

ishi

kaw

a on

Jun

e 10

, 201

6 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.2

016-

3969

-2 -1.5 -1 -0.5 0

hV

-6

-5

-4

-3

-2

-1L2error

ρ(FUN3D)u(FUN3D)v(FUN3D)w(FUN3D)p(FUN3D)ρ(FUN3D-i3rd)u(FUN3D-i3rd)v(FUN3D-i3rd)w(FUN3D-i3rd)p(FUN3D-i3rd)Slope 2Slope 3

Figure 9: Error convergence for Re∞ = 10.

-2 -1.5 -1 -0.5 0

hV

-6

-5

-4

-3

-2

-1

L2error

ρ(FUN3D)u(FUN3D)v(FUN3D)w(FUN3D)p(FUN3D)ρ(FUN3D-i3rd)u(FUN3D-i3rd)v(FUN3D-i3rd)w(FUN3D-i3rd)p(FUN3D-i3rd)Slope 2Slope 3

Figure 10: Error convergence for Re∞ = 100.

-2 -1.5 -1 -0.5 0

hV

-6

-5

-4

-3

-2

-1

L2error

ρ(FUN3D)u(FUN3D)v(FUN3D)w(FUN3D)p(FUN3D)ρ(FUN3D-i3rd)u(FUN3D-i3rd)v(FUN3D-i3rd)w(FUN3D-i3rd)p(FUN3D-i3rd)Slope 2Slope 3

Figure 11: Error convergence for Re∞ = 1000.

21 of 41

American Institute of Aeronautics and Astronautics

Dow

nloa

ded

by H

iroa

ki N

ishi

kaw

a on

Jun

e 10

, 201

6 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.2

016-

3969

(a) Surface grid. (b) Outer boundary grid.

Figure 12: The coarsest grid for the inviscid hemisphere cylinder case.

-1 -0.8 -0.6

hV

-4

-3.5

-3

-2.5

L∞

error

FUN3DFUN3D-i3rd(nB)-Q1FUN3D-i3rd(nexact)FUN3D-i3rd(nB)FUN3D-i3rd(nlumped)Slope 1Slope 2Slope 3

Figure 13: Entropy error convergence in L∞ norm.

22 of 41

American Institute of Aeronautics and Astronautics

Dow

nloa

ded

by H

iroa

ki N

ishi

kaw

a on

Jun

e 10

, 201

6 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.2

016-

3969

Figure 14: Irregular cube grid (253 nodes) for accu-racy verification study in Section V.B.1.

Figure 15: Slices of the irregular cube grid (253 nodes)for accuracy verification study in Section V.B.1.

-1.5 -1

hV

-5

-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

L2error

FUN3DFUN3D-i3rdHNS20-IIHNS20-IHNS20-I(Q)Slope 1Slope 2Slope 3

(a) Re∞ = 1.

-1.5 -1

hV

-6

-5

-4

-3

-2

-1L2error

FUN3DFUN3D-i3rdHNS20-IIHNS20-IHNS20-I(Q)Slope 1Slope 2Slope 3

(b) Re∞ = 10000.

Figure 16: Error convergence for the pressure p.

23 of 41

American Institute of Aeronautics and Astronautics

Dow

nloa

ded

by H

iroa

ki N

ishi

kaw

a on

Jun

e 10

, 201

6 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.2

016-

3969

-1.5 -1

hV

-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

L2error

FUN3DFUN3D-i3rdHNS20-IIHNS20-IHNS20-I(Q)Slope 1Slope 2Slope 3

(a) Re∞ = 1.

-1.5 -1

hV

-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

L2error

FUN3DFUN3D-i3rdHNS20-IIHNS20-IHNS20-I(Q)Slope 1Slope 2Slope 3

(b) Re∞ = 10000.

Figure 17: Error convergence for ∂xp.

-1.5 -1

hV

-2

-1.5

-1

-0.5

0

0.5

1

L2error

FUN3DFUN3D-i3rdHNS20-IIHNS20-IHNS20-I(Q)Slope 1

(a) Re∞ = 1.

-1.5 -1

hV

-2

-1.5

-1

-0.5

0

0.5

1

L2error

FUN3DFUN3D-i3rdHNS20-IIHNS20-IHNS20-I(Q)Slope 1

(b) Re∞ = 10000.

Figure 18: Error convergence for ∂xxp.

24 of 41

American Institute of Aeronautics and Astronautics

Dow

nloa

ded

by H

iroa

ki N

ishi

kaw

a on

Jun

e 10

, 201

6 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.2

016-

3969

Figure 19: The coarsest 3D tetrahedral grid used for the gradient accuracy test.Note that the z-coordinate is scaled by a factor of 20; the actual grid is still 10times smaller in the z direction.

Figure 20: Section of the 3D tetrahedral grid used for the gradient accuracy test.Note the different scalings in y and z axes.

25 of 41

American Institute of Aeronautics and Astronautics

Dow

nloa

ded

by H

iroa

ki N

ishi

kaw

a on

Jun

e 10

, 201

6 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.2

016-

3969

(a) Exact ∂zu. (b) FUN3D: Linear LSQ ∂zu.

(c) FUN3D-i3rd: Quadratic LSQ ∂zu. (d) HNS20-I: (gu)z/µv(= ∂zu).

(e) HNS20-I(Q): (gu)z/µv(= ∂zu). (f) HNS20-II: (gu)z/µv(= ∂zu).

Figure 21: Contours of the z-derivative of the velocity component u.

26 of 41

American Institute of Aeronautics and Astronautics

Dow

nloa

ded

by H

iroa

ki N

ishi

kaw

a on

Jun

e 10

, 201

6 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.2

016-

3969

Figure 22: Section plot at x = 0.0 for sphere (Grid1).

Figure 23: Surface grid for sphere (Grid1).

27 of 41

American Institute of Aeronautics and Astronautics

Dow

nloa

ded

by H

iroa

ki N

ishi

kaw

a on

Jun

e 10

, 201

6 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.2

016-

3969

(a) FUN3D. (b) FUN3D-i3rd.

(c) HNS20-I. (d) HNS20-I(Q).

Figure 24: Pressure contours over the sphere surface (Grid4).

28 of 41

American Institute of Aeronautics and Astronautics

Dow

nloa

ded

by H

iroa

ki N

ishi

kaw

a on

Jun

e 10

, 201

6 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.2

016-

3969

(a) FUN3D. (b) FUN3D-i3rd.

(c) HNS20-I. (d) HNS20-I(Q).

Figure 25: Pressure contours over the sphere surface and in the section at z = 0 (Grid4).

29 of 41

American Institute of Aeronautics and Astronautics

Dow

nloa

ded

by H

iroa

ki N

ishi

kaw

a on

Jun

e 10

, 201

6 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.2

016-

3969

(a) FUN3D: Linear LSQ ∂xv − ∂yu. (b) FUN3D-i3rd: Quadratic LSQ ∂xv − ∂yu.

(c) HNS20-I: [(gv)x − (gu)y]/µv(= ∂xv − ∂yu). (d) HNS20-I(Q): [(gv)x − (gu)y]/µv(= ∂xv − ∂yu).

Figure 26: Contours of the z-component of the vorticity on the sphere surface (Grid4).

30 of 41

American Institute of Aeronautics and Astronautics

Dow

nloa

ded

by H

iroa

ki N

ishi

kaw

a on

Jun

e 10

, 201

6 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.2

016-

3969

(a) FUN3D: Linear LSQ ∂xv − ∂yu. (b) FUN3D-i3rd: Quadratic LSQ ∂xv − ∂yu.

(c) HNS20-I: [(gv)x − (gu)y]/µv(= ∂xv − ∂yu). (d) HNS20-I(Q): [(gv)x − (gu)y]/µv(= ∂xv − ∂yu).

Figure 27: Contours of the z-component of the vorticity on the sphere surface and in the section at z = 0(Grid4).

31 of 41

American Institute of Aeronautics and Astronautics

Dow

nloa

ded

by H

iroa

ki N

ishi

kaw

a on

Jun

e 10

, 201

6 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.2

016-

3969

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

TotalD

ragCo

efficien

t

hv

Reynolds=101,Mach=0.15

FUN3DFUN3D-i3rdHNS20-IHNS20-I(Q)Experiment

(a) CD versus hV

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

0 0.01 0.02 0.03 0.04 0.05 0.06

TotalD

ragCo

efficien

t

h_DoF

Reynolds=101,Mach=0.15

FUN3DFUN3D-i3rdHNS20-IHNS20-I(Q)Experiment

(b) CD versus hDoF

Figure 28: Grid convergence of drag coefficient for sphere. CD is based on the projected area of the sphere:πd2/4, where d is the diameter of the sphere. The mesh spacing scale hV is the L1 norm of the cubic root of

the control volume; hDoF = (4N)−1/3

for the HNS schemes and hDoF = N−1/3 for others, where N is the totalnumber of nodes in the grid.

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Viscou

sDragCo

efficien

t

hv

Reynolds=101,Mach=0.15

FUN3DFUN3D-i3rdHNS20-IHNS20-I(Q)

(a) Pressure drag coefficient.

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Viscou

sDragCo

efficien

t

hv

Reynolds=101,Mach=0.15

FUN3DFUN3D-i3rdHNS20-IHNS20-I(Q)

(b) Viscous drag coefficient.

Figure 29: Grid convergence of pressure and viscous drag coefficients for sphere.

32 of 41

American Institute of Aeronautics and Astronautics

Dow

nloa

ded

by H

iroa

ki N

ishi

kaw

a on

Jun

e 10

, 201

6 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.2

016-

3969

Figure 30: 3D grid for Joukowsky airfoil (Grid1).

(a) Near-field view. (b) Near-field view close to the viscous wall.

Figure 31: Near-field views in (x, z)-plane for Joukowsky airfoil (Grid1).

33 of 41

American Institute of Aeronautics and Astronautics

Dow

nloa

ded

by H

iroa

ki N

ishi

kaw

a on

Jun

e 10

, 201

6 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.2

016-

3969

(a) FUN3D. (b) FUN3D-i3rd.

(c) HNS20-I. (d) HNS20-I(Q).

Figure 32: Pressure contours for Joukowsky airfoil (Grid1).

34 of 41

American Institute of Aeronautics and Astronautics

Dow

nloa

ded

by H

iroa

ki N

ishi

kaw

a on

Jun

e 10

, 201

6 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.2

016-

3969

(a) FUN3D: Linear LSQ |curlu|. (b) FUN3D-i3rd: Quadratic LSQ |curlu|.

(c) HNS20-I: |Egt|(= |curlu|). (d) HNS20-I(Q): |Egt|(= |curlu|).

Figure 33: Contours of the magnitude of the vorticity for Joukowsky airfoil (Grid1).

35 of 41

American Institute of Aeronautics and Astronautics

Dow

nloa

ded

by H

iroa

ki N

ishi

kaw

a on

Jun

e 10

, 201

6 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.2

016-

3969

(a) FUN3D: Linear LSQ applied twice ∂zzu. (b) FUN3D-i3rd: Quadratic LSQ applied twice ∂zzu.

(c) HNS20-I: Linear LSQ of gu - Equation (III.23). (d) HNS20-I(Q): Linear LSQ of gu - Equation (III.23).

Figure 34: Contours of the second-derivaitve ∂zzu for Joukowsky airfoil (Grid1).

36 of 41

American Institute of Aeronautics and Astronautics

Dow

nloa

ded

by H

iroa

ki N

ishi

kaw

a on

Jun

e 10

, 201

6 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.2

016-

3969

(a) FUN3D. (b) FUN3D-i3rd.

(c) HNS20-I. (d) HNS20-I(Q).

Figure 35: Pressure contours for Joukowsky airfoil (Grid2).

37 of 41

American Institute of Aeronautics and Astronautics

Dow

nloa

ded

by H

iroa

ki N

ishi

kaw

a on

Jun

e 10

, 201

6 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.2

016-

3969

(a) FUN3D: Linear LSQ |curlu|. (b) FUN3D-i3rd: Quadratic LSQ |curlu|.

(c) HNS20-I: |Egt|(= |curlu|). (d) HNS20-I(Q): |Egt|(= |curlu|).

Figure 36: Contours of the magnitude of the vorticity |ω| = |curlu| for Joukowsky airfoil (Grid2).

38 of 41

American Institute of Aeronautics and Astronautics

Dow

nloa

ded

by H

iroa

ki N

ishi

kaw

a on

Jun

e 10

, 201

6 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.2

016-

3969

(a) FUN3D: Linear LSQ applied twice ∂zzu. (b) FUN3D-i3rd: Quadratic LSQ applied twice ∂zzu.

(c) HNS20-I: Linear LSQ of gu - Equation (III.23). (d) HNS20-I(Q): Linear LSQ of gu - Equation (III.23).

Figure 37: Contours of the second-derivaitve ∂zzu for Joukowsky airfoil (Grid2).

39 of 41

American Institute of Aeronautics and Astronautics

Dow

nloa

ded

by H

iroa

ki N

ishi

kaw

a on

Jun

e 10

, 201

6 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.2

016-

3969

0.24

0.245

0.25

0.255

0.26

0.265

0.27

0.275

0.28

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

TotalD

ragCo

efficien

t

hv

Reynolds=1000,Mach=0.5,AoA=2degrees

FUN3DFUN3D-i3rdHNS20-IHNS20-I(Q)

(a) CD versus hV

0.24

0.245

0.25

0.255

0.26

0.265

0.27

0.275

0.28

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05

TotalD

ragCo

efficien

t

h_DoF

Reynolds=1000,Mach=0.5,AoA=2degrees

FUN3DFUN3D-i3rdHNS20-IHNS20-I(Q)

(b) CD versus hDoF

Figure 38: Grid convergence of drag coefficient for Joukowsky airfoil. CD is based on the chord length. The

mesh spacing scale hV is the L1 norm of the cubic root of the control volume; hDoF = (4N)−1/3

for the HNSschemes and hDoF = N−1/3 for others, where N is the total number of nodes in the grid. The FUN3D schemewas tested for one more fine grid with 3,487,040 nodes to confirm the drag coefficient convergence.

0.085

0.087

0.089

0.091

0.093

0.095

0.097

0.099

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

PressureDragCo

efficien

t

hv

Reynolds=1000,Mach=0.5,AoA=2degrees

FUN3DFUN3D-i3rdHNS20-IHNS20-I(Q)

(a) Pressure drag coefficient.

0.16

0.162

0.164

0.166

0.168

0.17

0.172

0.174

0.176

0.178

0.18

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Viscou

sDragCo

efficien

t

hv

Reynolds=1000,Mach=0.5,AoA=2degrees

FUN3DFUN3D-i3rdHNS20-IHNS20-I(Q)

(b) Viscous drag coefficient.

Figure 39: Grid convergence of pressure and viscous drag coefficients for Joukowsky airfoil.

40 of 41

American Institute of Aeronautics and Astronautics

Dow

nloa

ded

by H

iroa

ki N

ishi

kaw

a on

Jun

e 10

, 201

6 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.2

016-

3969

(a) 3D grid. (b) Section at y = 1.0.

Figure 40: Grid used for the 3D flat plate.

0.0011

0.00115

0.0012

0.00125

0.0013

0.00135

0.0014

0 0.01 0.02 0.03 0.04 0.05 0.06

DragCoe

fficien

t

hv

Reynolds=1million

Theory2DDragCoefficientFUN3DFUN3D-i3rdHNS20-IHNS20-I(Q)

(a) CD versus hV

0.0011

0.00115

0.0012

0.00125

0.0013

0.00135

0.0014

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

DragCoe

fficien

t

h_DoF

Reynolds=1million

Theory2DDragCoefficientFUN3DFUN3D-i3rdHNS20-IHNS20-I(Q)

(b) CD versus hDoF

Figure 41: Grid convergence of drag coefficient for 3D flat plate. The mesh spacing scale hV is the L1 norm

of the cubic root of the control volume; hDoF = (4N)−1/3

for the HNS schemes and hDoF = N−1/3 for others,where N is the total number of nodes in the grid.

41 of 41

American Institute of Aeronautics and Astronautics

Dow

nloa

ded

by H

iroa

ki N

ishi

kaw

a on

Jun

e 10

, 201

6 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.2

016-

3969