Incompressible uid solver in LS-DYNAftp.lstc.com/anonymous/outgoing/inaki/docs/pdf_icfd/ICFD...Test Case Name Incompressible uid solver in LS-DYNA Test Case ID ICFD-THE-1.1 Test Case

ICFD THEORY MANUAL

Incompressible fluid solver

in LS-DYNA

Tested with LS-DYNA R© R7.0 Revision Beta

Wednesday 8th October, 2014

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LSTC-LS-DYNA-ICFD-THE-1.1-1 i

Document Information

Confidentiality external

Document Identifier LSTC-LS-DYNA-ICFD-THE-1.1-1

Prepared by LSTC R© Inaki Caldichoury, LSTC R© Rodrigo R. Paz

Approved by LSTC R© Facundo Del Pin, LSTC R© Rodrigo R. Paz

Number of pages 61

Date created Wednesday 8th October, 2014

Distribution External

ii LSTC-LS-DYNA-ICFD-THE-1.1-1

Contents

1 Introduction 11.1 Purpose of this Document . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Document Information 2

3 Notations and physical variables 3

4 The ICFD solver 44.1 Fluid Mechanics equations . . . . . . . . . . . . . . . . . . . . . . . . 4

4.1.1 The incompressibility condition . . . . . . . . . . . . . . . . . 44.1.2 Governing set of equations . . . . . . . . . . . . . . . . . . . . 44.1.3 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . 54.1.4 ALE approach for mesh movement . . . . . . . . . . . . . . . 6

4.2 The Fractional Step method . . . . . . . . . . . . . . . . . . . . . . . 74.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74.2.2 Splitting the Momentum Equations . . . . . . . . . . . . . . . 74.2.3 Equation of Incompressibility . . . . . . . . . . . . . . . . . . 94.2.4 Three Step Fractional Method . . . . . . . . . . . . . . . . . . 9

4.3 Spatial discretization by the Finite Element Method . . . . . . . . . . 104.3.1 The FEM system . . . . . . . . . . . . . . . . . . . . . . . . . 104.3.2 Predictor corrector scheme . . . . . . . . . . . . . . . . . . . . 13

4.4 Integration scheme stabilization . . . . . . . . . . . . . . . . . . . . . 144.4.1 Pressure stabilization . . . . . . . . . . . . . . . . . . . . . . . 144.4.2 Convection stabilization . . . . . . . . . . . . . . . . . . . . . 15

4.5 Watching and interpreting the analysis . . . . . . . . . . . . . . . . . 164.6 Non-Inertial Reference Frame . . . . . . . . . . . . . . . . . . . . . . 17

5 Structure and Thermal coupling 185.1 Fluid Structure Interaction (FSI) . . . . . . . . . . . . . . . . . . . . 18

5.1.1 Types of FSI coupling . . . . . . . . . . . . . . . . . . . . . . 185.1.2 Partitioned Approximation for Strong FSI Coupling . . . . . . 195.1.3 Evaluation of the Laplace Matrix for FSI problems . . . . . . 205.1.4 FSI resolution steps . . . . . . . . . . . . . . . . . . . . . . . . 215.1.5 Watching and controlling the analysis for FSI cases . . . . . . 24

5.2 Thermal coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255.2.2 The heat equation . . . . . . . . . . . . . . . . . . . . . . . . 25

LSTC-LS-DYNA-ICFD-THE-1.1-1 iii

5.2.3 Spatial discretization . . . . . . . . . . . . . . . . . . . . . . . 255.2.4 Coupling with the thermal solver . . . . . . . . . . . . . . . . 265.2.5 Watching and controlling the analysis for thermal analysis. . . 27

5.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

6 Free surface and mutliphase handling 286.1 The level set method . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

6.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286.1.2 The level set function . . . . . . . . . . . . . . . . . . . . . . . 286.1.3 The convection equation . . . . . . . . . . . . . . . . . . . . . 296.1.4 Numerical integration . . . . . . . . . . . . . . . . . . . . . . . 29

7 Anistropic/Isotropic Generalized Flow Through Porous Media 327.1 Generalized Porous Media Model: isotropic media . . . . . . . . . . . 32

7.1.1 Model Parameters limits . . . . . . . . . . . . . . . . . . . . . 337.2 Anisotropic Generalized Flow Through Porous Media. . . . . . . . . . 337.3 Heat Transfer in Porous Media: coupled PM/NS heat transfer. . . . . 337.4 Computing porous parameters using the pressure-velocity (p− u) ex-

perimental data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

8 Turbulence models 378.1 The RANS model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

8.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378.1.2 The Reynolds Averaged Equations . . . . . . . . . . . . . . . 378.1.3 The k-ε model . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

8.2 The LES model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398.2.2 LES equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 398.2.3 The Smagorinsky model . . . . . . . . . . . . . . . . . . . . . 40

8.3 Wall function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408.4 Turbulence synthesis method for Large Eddy Simulations . . . . . . . 418.5 The Random Flow Generator in Python: How Inlet turbulence gen-

erator works (basically) in ICFD/LS-DYNA . . . . . . . . . . . . . . 428.6 Smirnov’s algorithm to generate inhomogeneous anisotropic turbu-

lence at inflow boundaries in LES runs . . . . . . . . . . . . . . . . . 43

9 The Volume Mesher 509.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509.2 The Delaunay criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

iv LSTC-LS-DYNA-ICFD-THE-1.1-1

9.3 Initial Volume mesh building steps . . . . . . . . . . . . . . . . . . . 509.4 Local refinement tools . . . . . . . . . . . . . . . . . . . . . . . . . . 51

9.4.1 Imposing the mesh size on a volume domain . . . . . . . . . . 519.4.2 The boundary layer mesh . . . . . . . . . . . . . . . . . . . . 53

9.5 Volume mesh remeshing criteria . . . . . . . . . . . . . . . . . . . . . 539.5.1 Mesh deformation criteria . . . . . . . . . . . . . . . . . . . . 539.5.2 Using adaptivity . . . . . . . . . . . . . . . . . . . . . . . . . 53

9.6 Watching the mesh quality . . . . . . . . . . . . . . . . . . . . . . . . 559.6.1 Mesh quality output file . . . . . . . . . . . . . . . . . . . . . 559.6.2 Terminal information . . . . . . . . . . . . . . . . . . . . . . . 56

A Consistent units for ICFD 61

LSTC-LS-DYNA-ICFD-THE-1.1-1 v

1 Introduction

1.1 Purpose of this Document

A detailed description of the analytical equations solved by the incompressible fluidsolver is given as well as the numerical methods used. Additional descriptions ofthe equations used for some of the main features of the solver is also provided. Theobjective of this document is to offer the reader who wishes to understand and usethe incompressible fluid solver of LS-DYNA a precise and easy way to understandinsight of the formulas, notions and theories employed by the solver.

LSTC-LS-DYNA-ICFD-THE-1.1-1 1

2 Document Information

Test Case Summary

Confidentiality external use

Test Case Name Incompressible fluid solver in LS-DYNA

Test Case ID ICFD-THE-1.1

Test Case Status active

Test Case Classification Theory document

Metadata Theory

Table 1: Test Case Summary

2 LSTC-LS-DYNA-ICFD-THE-1.1-1

3 Notations and physical variables

Table 2 provides the meaning of each symbol and the SI unit of measure while table3 provides the conventions and operators used :

Symbol : Meaning : Units (SI) :

~u fluid velocity meters per second

p fluid pressure Pascals

ρ fluid density kilograms per cubic meter

µ fluid dynamic viscosity Pascals seconds

ν fluid kinematic viscosity square meters per second

α thermal diffusivity square meters per second

Table 2: Variables and constants

grad, curl, div operators ~∇, ~∇×, ∇·

vectors in R3 arrow overhead : ~vec

matrices bold : P

vectors in FEM/BEM systems small : ai

Table 3: Operators and conventions


4 The ICFD solver

4.1 Fluid Mechanics equations

4.1.1 The incompressibility condition

The present solver aims to solve the incompressible Navier-Stokes equations. Tradi-tionally, a fluid may be considered incompressible when the Mach number is lowerthan 0.3, i.e

M =V

a≤ 0.3, (1)

where V is the velocity of the flow relative to a fixed object and a is the speed ofsound in the medium (in air, we thus have v ≤ 370 km/h).In fluid dynamics, the continuity equation states that, in any steady state process,the rate at which mass enters a system is equal to the rate at which mass leaves thesystem. The continuity equation is analogous to Kirchhoff’s current law in electriccircuits.The differential form of the continuity equation is

∂ρ

∂t+∇ · (ρ~u) = 0. (2)

If ρ is a constant, as in the case of incompressible flow, the mass continuity equationsimplifies to a volume continuity equation

∇ · ~u = 0. (3)

4.1.2 Governing set of equations

Conservation of momentum and mass for incompressible Newtonian fluids in theEulerian conventional form are represented by the Navier-Stokes equations combinedwith the continuity equations as follows:

ρ

(duidt

+ uj∂ui∂xj

)=

∂σi,j∂xj

+ ρfi in Ω, (4)

∂ui∂xi

= 0 in Ω. (5)

The total stress tensor is given by

σij = −pδij + µ

(∂ui∂xj

+∂uj∂xi− 2

3

∂ul∂xl

δij

). (6)


For near incompressible flows we have that

∂ui∂xi ∂ui

∂xj, (7)

and the last term in the brackets may be neglected from Equation (6). Then,

σij ≈ −pδij + µ

(∂ui∂xj

+∂uj∂xi

). (8)

In the same way, the term ∂σij/∂xj in the momentum equation may be simplifiedfor near incompressible flows as

∂σij∂xj

= − ∂p

∂xjδij +

∂

∂xj

[µ

(∂ui∂xj

+∂uj∂xi

)]= − ∂p

∂xjδij + µ

∂

∂xj

(∂ui∂xj

)+ µ

∂

∂xj

(∂uj∂xi

)= − ∂p

∂xjδij + µ

∂

∂xj

(∂ui∂xj

)+ µ

∂

∂xi

(∂uj∂xj

)≈ − ∂p

∂xjδij + µ

∂

∂xj

(∂ui∂xj

).

(9)

The above expression simplifies the viscous term so that we end up with the followingsystem

ρ

(∂ui∂t

+ uj∂ui∂xj

)= − ∂p

∂xi+ µ

∂2ui∂xj∂xj

+ ρfi in Ω (10)

∂ui∂xi

= 0 in Ω (11)

4.1.3 Boundary conditions

The set of differential equations presented above is incomplete if the appropriate setof boundary conditions and initial conditions is not specified. For the continuumproblem conditions specifying velocities and stress have to be imposed.The components of the velocity vector will be replaced, according to the constraintsrelated to the boundary, by the proper imposed value. Over a rigid wall the boundaryconditions follow directly from the momentum equations. For a free-slip wall thenormal velocity must vanish; for a non-slip wall the tangential components must, inaddition, vanish. These conditions in general can be defined as

ui = vi on Γv, (12)


where vi indicates the function that is imposed on the boundary.The other type of boundary conditions are imposed on the free surface. These arethe most difficult to identify as the shape of the free surface has to be computedahead. The principles that form the basis for the free surface boundary conditionsare stated as follow

• stress tangential to the surface must vanish,

• stress normal to the surface must exactly balance any externally applied normalstress.

In this case these conditions are expressed as

ti = σijnj = ti on Γf , (13)

where nj is the direction cosine of the outward normal on the boundary with respectto the xj axis. Both surfaces Γv and Γf are two disjoint non-overlapping subsets ofthe boundary Γ.The set of initial conditions for the Lagrangian Navier-Stokes problem will be givenspecifying the velocity and pressure at the initial time

vi(xi, 0) = v0i (xi), (14)

p(xi, 0) = p0(xi), (15)

where the initial velocity v0i has to satisfy the incompressibility constrain ∂vi/∂xi = 0.

4.1.4 ALE approach for mesh movement

In a CFD only analysis, the moving reference frame is fixed in space and a fullEulerian formulation is achieved. However, in cases of Fluid Structure Interaction(FSI) problems, the boundaries between the solid and the fluid are Lagrangian anddeform with the structure. An ALE formulation is hence retrieved. This approachallows a strong and exact imposition of the solid boundary conditions on the fluid.The solid and fluid geometry must match at the interface but not necessarily themeshes. Rewriting the equations of motion for an ALE formulation, we now have

ρ

(∂ui∂t

+ (uj − vj)∂ui∂xj

)= − ∂p

∂xi+ µ

∂2ui∂xj∂xj

+ ρfi in Ω (16)

∂ui∂xi

= 0 in Ω (17)

where vj is the velocity of the moving frame.


4.2 The Fractional Step method

4.2.1 Introduction

For the time integration of the Navier-Stokes equation a projection method wasadopted ([6], [7], [27]).These methods stand out from classical monolithic schemesas pressure and velocity are uncoupled. Thus four linear systems of equations areobtained, namely three for the momentum equation and one to solve the incompress-ibility constrain.The present scheme consists of three steps until the final velocity is achieved ([25]).In the first step, a predictor velocity u∗i is computed. This velocity does not satisfythe incompressibility constrain given by Equation (5). Thus, in the second step thevelocity ui is projected into a space of divergence free vector field and a Poissonequation of pressure is obtained. This pressure will correct ui to get a final stepdivergence free velocity un+1

i . The final step consists of moving the particles to theirnew time step position xn+1 . If this is done iteratively then convergence is achievedwhen the particles always move to the same final position. This class of schemes arenow widely used in practice and have been rigously analyzed in ([20], [12], [16], [30],[32], [10], [8]).

4.2.2 Splitting the Momentum Equations

Let us start with the time derivative that is approximated with backward differencesof second order as

duidt≈ 1.5un+1

i − 2uni + 0.5un−1i

∆t(18)

However, for clarity purposes, the following demonstrations will use simple forwarddifferences approximation keeping in mind that the procedure is similar with back-ward differences approximation

duidt≈ un+1

i − uni∆t

(19)

If we now consider f a generic function of time and fn the value of f at tn = n∆t oran approximation of it, and let fn+θ = θfn+1 + (1 − θ)fn, we can introduce a newvariable on the left side of Equation (10) and add and subtract a pressure term onthe right hand side as follows

ρun+1i − u?i + u?i − uni

∆t=

∂

∂xi

[−pn+1 + γpn − γpn

]δij +

[µ∂

∂xj

(∂ui∂xj

)− ρuj

∂ui∂xj

+ ρfi

]n+θ

,(20)


where u?i are fictitious variables called fractional velocities and γ is a scalar valuethat varies from 0 to 1 and that will be explored later. The value of θ = 0 impliesan explicit forward Euler scheme, θ = 1 is an implicit backward Euler scheme andθ = 0.5 is the Crank-Nicholson second order semi-implicit scheme. In the presentanalysis θ = 1 will be adopted. We can see that the pressure term has been taken outfrom the implicit-explicit term [·]n+θ and only the implicit part is taken into account.Rewriting Equation (20), and reordering some terms, we get

un+1i − u?i + u?i =

∆t

ρ

∂

∂xi

[−pn+1 + γpn

]δij + uni−

γ∆t

ρ

∂pn

∂xjδij +

∆t

ρ

[µ∂

∂xj

(∂ui∂xj

)+ ρfi − ρuj

∂ui∂xj

]n+θ

,

(21)

The last three terms of Equation (21) will be related to u?i in such a way that wemay write

u?i = uni − γ∆t

ρ

∂pn

∂xiδij +

∆t

ρµ∂

∂xj

(∂un+θ

i

∂xj

)−∆tun+θ

j

∂un+θi

∂xj+ ∆t fi, (22)

un+1i = u?i +

∆t

ρ

∂

∂xi

[−pn+1 + γpn

]δij, (23)

in which fi is considered constant in time and pn is the pressure field at time tn butevaluated at the final position. In the viscous and advection terms, the values ofun+1i and un+1

j will be estimated using u?i . The term un+1i will be estimated as un+1

i .

The choice of un+1i will be explicited later. This allows to write Equation (22) as

u?i −∆t

ρµθ

∂

∂xj

(∂u?i∂xj

)+ ∆tu?j

∂un+1i

∂xj= uni − γ

∆t

ρ

∂pn

∂xiδij+

∆t

ρµ(1− θ) ∂

∂xj

(∂uni∂xj

)−∆t(1− θ)unj

∂uni∂xj

+ ∆t fi,

(24)

which is the actual expression to compute the predictor velocity. The parameter γintroduced earlier, determines the amount of pressure splitting. Thus when γ = 1it means small pressure splitting and γ = 0 implies the largest pressure splitting.When γ = 0 a first order scheme in time is obtained. On the contrary when γ = 1a second order scheme is found. A second order scheme gives unstable values of thepressure field that need to be corrected. The stabilization technique used in this casewill be introduced later.


4.2.3 Equation of Incompressibility

The main objective of this phase is to obtain a pressure field that, when replaced inEquation (23), satisfies the incompressibility constraint. Once the intermediate ve-locity u?i has been computed from Equation (24), the end of step velocity is obtainedby adding to u?i the dynamical effect of the still unknown pressure pn+1, which mustbe determined such that the incompressibility constraint from Equation (5) remainsatisfied.Taking Equation (23) and applying the divergence operator on both sides

∂

∂xi(un+1

i − u?i ) =∆t

ρ

∂

∂xi

∂

∂xi

[−pn+1 + γpn

]δij. (25)

By means of Equation (5) the first term on the left hand side vanishes and we maywrite

∂

∂xi(−u?i ) =

∆t

ρ

∂

∂xi

∂

∂xi

[−pn+1 + γpn

]δij. (26)

which is a linear elliptic equation of pressure. Solving Equation (26) with the appro-priate boundary conditions will determine the new time step pressure.

4.2.4 Three Step Fractional Method

Each iteration inside the time interval ∆t will involve to compute three sets of equa-tions, each of them related to different steps of the time integration scheme, namelyusing θ = 1.u?i :

u?i −∆t

ρµ∂

∂xj

(∂u?i∂xj

)+ ∆tu?j

∂un+1i

∂xj= uni − γ

∆t

ρ

∂pn

∂xiδij + ∆t fi, (27)

pn+1:∂

∂xi(−u?i ) =

∆t

ρ

∂

∂xi

∂

∂xi

[−pn+1 + γpn

]δij, (28)

un+1i :

un+1i = u?i +

∆t

ρ

∂

∂xj

[−pn+1 + γpn

]δij. (29)


4.3 Spatial discretization by the Finite Element Method

4.3.1 The FEM system

Let us first introduce the weak form of the equations of motion that have been alreadydecoupled in Equations (27), (28) and (29). We will use indistinctly the letter N toidentify either trial functions or interpolating functions. Multiplying the equationsby the trial function N and integrating over the volume Ω:u?i : ∫

Ω

Ni u?i dΩ−∆t

∫Ω

Niµ

ρ

∂

∂xj

(∂u?i∂xj

)dΩ +

∫Ω

∆tNiu?j

∂un+1i

∂xjdΩ =

∫Ω

Ni uni dΩ− γ∆t

∫Ω

Ni

ρ

∂pn

∂xiδijdΩ + ∆t

∫Ω

Ni fidΩ,

(30)

pn+1: ∫Ω

N∂

∂xi(−u?i )dΩ = ∆t

∫Ω

N

ρ

∂

∂xi

∂

∂xi

[−pn+1 + γpn

]δijdΩ, (31)

un+1i : ∫

Ω

Ni un+1i dΩ =

∫Ω

Ni u?i dΩ + ∆t

∫Ω

Ni

ρ

∂

∂xi

[−pn+1 + γpn

]δijdΩ. (32)

where the parameters ∆t and γ are considered constant during the integration.Integrating by parts the stress term in Equation (30) and the pressure term in (31),use of the divergence theorem and the boundary conditions of Equation (13) and(Equation 12) let us arrive to the following set of equations:u?i : ∫

Ω

Ni u?i dΩ + ∆t

∫Ω

µ

ρ

∂Ni

∂xj

(∂u?i∂xj

)dΩ +

∫Ω

∆tNiu?j

∂un+1i

∂xjdΩ =

∫Ω

Ni uni dΩ− γ∆t

∫Ω

Ni

ρ

∂pn

∂xiδijdΩ + ∆t

∫Ω

Ni fidΩ +

∫Γ

Ni

[∂u?i∂xj

nj

]Γf

dΓ,

(33)

pn+1: ∫Ω

∂N

∂xiu?i dΩ = ∆t

∫Ω

1

ρ

∂N

∂xi

∂

∂xi

[pn+1dΩ− γpn

]δij+∫

Γ

Ni

[u?i +

∆t

ρ

∂

∂xi(−pn+1 + γpn)δij

]Γv

nidΓ,

(34)


un+1i : ∫

Ω

Ni un+1i dΩ =

∫Ω

Ni u?i dΩ + ∆t

∫Ω

Ni

ρ

∂

∂xi

[−pn+1 + γpn

]δijdΩ, (35)

where ni is the outward normal vector to the Γv boundary. Comparing the boundaryterm in Equation (34) and the end of step velocity (Equation (35)) we may write:∫

Γ

N

[u?i +

∆t

ρ

∂

∂xi(−pn+1 + γpn)δij

]Γv

nidΓ =

∫Γ

N [un+1i ]ΓvnidΓ. (36)

To complete this set of equations we need to specify the essential and natural bound-ary conditions for Equation (33) and Equation (34). In the first case the derivativeof the velocity normal to the free surface will vanish resulting in the following naturalboundary condition [

∂u?i∂xj

nj

]Γf

= 0. (37)

Also free-slip or no-slip velocity conditions over the walls may be imposed. In thesecond case there will be two sets of boundary conditions namely essentials over Γfand Γv, we have

p = 0 on Γf , (38)

un+1i = un+1

i on Γv, (39)

which means that the value of pressure will be imposed over the free surface and thenormal velocity of the fluid over particles with a given known velocity will take theknown normal velocity of the particles.The full version of the discrete problem takes place when the unknown fields areapproximated over the finite elements. Then within an element, the velocity andpressure will be represented by

ui = Nluli, (40)

p = Nlpl, (41)

where l represents a node of the element. Replacing Equation (40) and Equation(41) in Equation (33) and (35) a final set of equations in a more compact form isobtained


ρMu?i + ∆tK(µ)u?i + S(ρun+1i )u?j = ρMuni − γ∆tGpn + ∆tF, (42)

∆tL

(1

ρ

)pn+1 = Du?i + γ∆tL

(1

ρ

)pn − U, (43)

ρMun+1i = ρMu?i −∆tG(pn+1 − γpn), (44)

where the matrices are defined by:

M =

∫Ω

Nn+1q Nn+1

l dΩ, (45)

K(µ) =

∫Ω

(µ∂Nn+1

q

∂xi

∂Nn+1l

∂xi

)dΩ, (46)

D =

∫Ω

(∂Nn+1

q

∂xiNn+1l

)dΩ, (47)

G =

∫Ω

(Nn+1q

∂Nn+1l

∂xi

)dΩ, (48)

1

ρL =

∫Ω

(1

ρ

∂Nn+1q

∂xi

∂Nn+1l

∂xi

)dΩ, (49)

F =

∫Ω

Nn+1q fidΩ, (50)

S(ρui) =

∫Ω

(ρNqu

n+1i

∂Nl

∂xi

)dΩ, (51)

U =

∫Γ

Nn+1q un+1

i nidΓ. (52)

The indices (q, l) denote the nodes of an element and n+ 1 implies that the interpo-lating functions have been computed at time step tn+1. The matrix M in Equation(44) is actually the lumped mass matrix M of Equation (45) obtained by adding theelements of M over a row and placing the result on the diagonal of the matrix.

Equations (42),(43) and (44) form the system that needs to be solved.


4.3.2 Predictor corrector scheme

In the previously defined system of equations it can be observed that in order to beable to solve Equation (42), an estimation of un+1

i needs to be given. The choice ofun+1i can be the object of discussions as it needs to be the closest possible to the

actual value of un+1i . An estimated average value of : un+1

i = 2uni − un−1i is being

used by default for the first solve of the system. However, the solver also provides anoption to trigger an iterative procedure on the system. Once u∗ and pn+1 have beendetermined using Equation (42) and (43), they can be re-injected in the system inorder to form an iterative loop

ρMu?k + ∆tK(µ)u?k + S(ρu?k−1)u?k = ρMun − γ∆tGpn+1k−1 + ∆tF, (53)

∆tL

(1

ρ

)pn+1k = Du?k + γ∆tL

(1

ρ

)pn − U, (54)

where k is the number of iterations (the vector indices do not show for clarity pur-poses).This procedure aims to bring more precision to the final pressure and velocity valuesbut is often very time consuming. It must be therefore used with caution in specificcases. For stability purposes, this method is authomatically used for the first ICFDtime step where iterations are conducted. Otherwise, it can be accessed through theICFD CONTROL SPLIT card (see Keyword manual).


4.4 Integration scheme stabilization

4.4.1 Pressure stabilization

The system formed by Equations (42), (43) and (44) can be solved as such but itcan be shown that stability issues arise [8] depending on the time step size. It can beanticipated that if ∆t is very small, stability problems may occur, especially for thesecond order scheme γ = 1 used by the solver. The orthogonal subscale stabilizationpresently used has been studied by [9].Let’s start by observing that the DM−1G represents the Laplacian operator at thecontinuous level. When using discrete elements, this leads to the approximation

DM−1G ' L, (55)

and let us now introduce the auxiliary variable Π such as

Π = M−1Gp. (56)

This will allow us to introduce a diffusion term in order to stabilize the pressure termthat will not lower the order of the scheme,

ρMu?i + K(µ)u?i + S(ρun+1i )u?j = ρMuni − γ∆tGpn + ∆tF, (57)

∆tL

(1

ρ


(1

ρ

)pn − U− τ(Lpn+1 −Dπn),

(58)


Π = M−1Gp, (60)

where τ is a stabilization parameter which depends on the local element sizes andbehaves as

τ ≤ Ch2

ν, (61)

where C is a constant.This stabilization term is consistent with the approximation introduced by Equation(55) which means that the stabilization term will get smaller when using a finer meshi.e when getting closer to the hypothetical continuous level.


4.4.2 Convection stabilization

The idea behind the stabilization of the convection term is similar to the pressurestabilization and has been developed in [9]. It consists of adding a low order stabi-lization term and recover the higher order of the advection operator by subtractinga higher order term. This term is built by projecting the velocity orthogonal to thefinite element space. We will only give the final system here that will consist of

ρMu?i + ∆tK(µ)u?i + S(ρun+1i )u?j = τ(Sy(u

ni )ynj − Su(u

ni )un+1

j ) + ρMuni − γ∆tGpn + ∆tF,

(62)

Myni −C(u?i )u?j = 0, (63)

∆tL

(1

ρ


(1

ρ

)pn − U− τ(Lpn −Dπn),

(64)


Π = M−1GP, (66)

with

Su =

∫Ω

(∂Nq

∂xiuni∂Nl

∂xiuni

)dΩ, (67)

Sy =

∫Ω

(∂Nq

∂xiuniNl

)dΩ, (68)

C =

∫Ω

(Nqu

ni

∂Nl

∂xi

)dΩ, (69)


4.5 Watching and interpreting the analysis

We have seen in the previous sections that the fractional step method consists insegregating the velocity and pressure terms thus resulting in Equation (42) and (43)that must be solved implicitly before finally updating the velocity using Equation(44). When running a CFD only analysis, it is possible for the user to track thesolving of these systems through the terminal window. Assuming that the maximumlevel of output information defined in the ICFD CONTROL OUTPUT card has beentriggered, this kind of information extracted from a 2D case, will appear:

21 t 4.6000E-01 dt 2.00E-02 ICFD time stepICFD::Fluid → Solving...PFEM:: It 2PFEM:: It 2ICFD:: Sub-step Res: 0.000203344 0.000001065PFEM:: It 5Frac Step Pres Residual = 0.016417PFEM:: It 2PFEM:: It 2ICFD:: Sub-step Res: 0.000079993 0.000000419ICFD::Saving Results → Saving...22 t 4.8000E-01 dt 2.00E-02 ICFD time step

After each ICFD time step, the solver will give the number of iteration that wereneeded in order to solve Equation (42) in the three directions of velocity (two in2D cases) as well as the associated residual. The solver will then proceed to solveEquation (43) and will also output the number of iterations as well as the associatedresidual. Finally, the solver will give the number of iterations needed in order tosolve Equation (44) in the three directions (two in 2D cases) as well as the associatedresidual. The solver will then proceed to the next ICFD time step. This way, theuser can make sure that the residuals stay low at each time step and that the analysisis not diverging.


4.6 Non-Inertial Reference Frame

The Navier-Stokes equation for incompressible flows may be modified to allow thesimulation in a moving reference frame. The reference frame considered here isrotating at a constant speed and thus it is undergoing acceleration with respectto the stationary frame. The rotating frame is then called a non-inertial referenceframe. This feature is useful in application such as wind turbines, turbomachineryand every other rotating problem that admits a steady state solution.To account for the rotating velocity the fluid velocity is transformed

va = vr + vΩ,

withvΩ = Ω× r

where va is the absolute fluid velocity (the velocity viewed from the stationary frame),vr is the relative velocity (the velocity viewed from the rotating frame) and vΩ is thevelocity of the rotating frame, Ω is the angular velocity (considered constant in thisanalysis) and r is the vector from the axis of rotation.Apart from the velocity transformations a rotating frame induce forces due to theCoriolis and centrifugal accelerations which are given by −2Ω × v and −Ω × Ω × rrespectively.Depending where the observer is placed the Navier-Stokes equations may be writtenin a relative form or in absolute form.For the absolute form:

ρ(∂v

∂t+ (v − vΩ)

∂v

∂xj) = − ∂p

∂xi+ µ

∂v

∂x2j

− ρ(Ωi × v) + Fi

In the relative reference frame the equations are re-written in terms of vr

ρ(∂vr∂t

+ vr∂vr∂xj

) = − ∂p

∂xi+ µ

∂v

∂x2j

− ρ(2Ωi × vr + Ωi × Ωi × ri) + Fi

The equations above are valid for steady state solutions within the rotating frame.Nevertheless these equations can also be applied to unsteady problems with rea-sonable results. The equation expressed in absolute reference frame is of particularinterest since it allows the definition of rotating sub-domains.


5 Structure and Thermal coupling

5.1 Fluid Structure Interaction (FSI)

5.1.1 Types of FSI coupling

Fluid-structure interaction problems involving an incompressible viscous flow andelastic non linear-structure have been solved in the past using different methods.

The monolithic approach considers the fluid and the solid as a single domain withthe fluid and solid equations solved together in a coupled way. However, solving thepressure together with the rest of the unknowns (typically velocities or displacements)is too expensive from the computational point of view : the non-linear system to besolved is large and ill-conditioned with always non-defined positive matrices.

A second approach would be to segregate the pressure from the velocity in the mono-lithic scheme. This would still imply that the fluid and solid equations are solved ina coupled way in the same system and would therefore require an implementationof the solid equations directly in the ICFD solver. This would not be a practicalsolution and would contradict the objective of the present solver which is to makefully use of LS-DYNA’s mechanical solver capabilities in order to solve complex fluidstructure interaction problems.

A third approach would be by using a partitioned (or staggered) method ([15], [13],[26], [22]) where the fluid and solid equations are uncoupled and that therefore allowsusing specifically designed codes on the different domains and offer significant benefitsin terms of efficiency: smaller and better conditioned subsystems are solved insteadof a single problem. It is the method adopted by the present solver in order to solveFSI problems.

In the partitioned approach, two schemes are distinguished : loosely (or weakly) ([21])coupled scheme or strongly coupled scheme ([28], [29], [31], [33], [14], [35], [11]). Bothare available when using the ICFD solver in LS-DYNA. Loosely coupled schemesrequire only one solution of either field per time step in a sequentially staggeredmanner and are thus particularly appealing in terms of efficiency. However, theytend to become unstable when the “added-mass effect” is significant([5], [3]).

In fluid mechanics, added mass or virtual mass is the inertia added to a systembecause an accelerating or decelerating body must move some volume of surroundingfluid as it moves through it, since the object and fluid cannot occupy the samephysical space simultaneously. However the name “added-mass effect” has beenused in the literature to indicate the numerical instabilities that typically occur inthe internal flow of an incompressible fluid whose density is close to the structuredensity. Other causes for instabilities include the elasticity coefficients or a time


step too small. The added mass effect therefore does usually not occur in aero-elasticity problems as the solid’s density is a lot higher than the fluid’s density, butit becomes very important in several other applications such as bio-mechanics wherethe materials are normally muscles and arteries and the fluid is blood.

Strongly coupled schemes require the convergence of the fluid and solid variables atthe interface and give, after an iterative process, the same results as non-partitionedschemes. However, they are also subject to the added mass effect resulting in anon convergence of the solution. Special stabilization techniques must therefore bedeveloped in order to diminish its influence and aim for a wider range of applicationsfor fluid structure interactions. This stabilization method will be developed in thenext section. Figure (1) gives a summary of the different FSI couplings possible,those implemented in the ICFD solver and the influence of the added mass effect.

Figure 1: Summary of the different FSI couplings and the numerical troubles inducedby the added-mass effect.

5.1.2 Partitioned Approximation for Strong FSI Coupling

The approach implemented in LS-DYNA to obtain a stronger FSI coupling can befound in [17]. It relays on finding an approximation of the tangent operator at theFSI interface without having to do a monolithic assembly. The contribution of thesolid elements on the interface is added to the fluid elements when the pressure


Laplace equation is built. It can be shown that this procedure greatly improves theconvergence of the FSI coupling. It should be noted that this is still a partitionedapproach and convergence may still be challenging for some kind of problems.

5.1.3 Evaluation of the Laplace Matrix for FSI problems

A brief introduction to the work in [17] will be presented with some details on theconstruction of the interface Laplace matrix.Lets assume that the Laplace matrix may be split into two parts, one for the fluiddomain and one for the fluid-solid interface as follows:

Lτ ≈ Lτf + Lτs , (70)

where Lτf is the Laplace matrix defined in the fluid domain including the interfaces

Lτf =∑

(τfLe), (71)

with Le being the Laplace matrix for the element and

τf =

(ρ

∆t+µ

h2+‖V ‖h

)−1

, (72)

Lτs is a Laplace matrix corresponding only to the fluid solid interface

Lτs =∑

(τsLe), (73)

where for a hyper-elastic solids the value of τs may be defined as

τs =

(ρsolid∆t

+∆tλ

Jh2+

∆tG

Jh2

)−1

, (74)

Le

=

[∫Ω

∂Nsf

∂xj

∂Nsf

∂xjdΩ

]s

(75)

where λ and G are Lame parameters, J the Jacobian matrix and Le

is the Laplacematrix of the solid elements evaluated only with the shape function Nsf that aredifferent from zero on the fluid-structure interface.Equation (70) may also be written as

Lτ ≈ Lτf + βLτf , (76)

with


β =τsτf, (77)

and

Lτf =∑

(τf Le), (78)

This means that the Laplace interface matrix Lτf may be neglected for small valuesof the β parameter. This is for instance the case when ρs >> ρf and the added-masseffect is not present. However, for other physical properties, the β parameter maynot be negligible and the Laplace interface matrix must be evaluated in order toobtain good results. It has been shown ([17]) that the βLtauf term improves theconvergence when ρf ≈ ρs, small time steps and when using soft materials.

5.1.4 FSI resolution steps

In section 5.1.1, we have classified the fluid structure interaction types and seenthat the ICFD solver allows loosely or strongly coupled schemes. Section 5.1.3 hasshowed the stabilization steps adopted in order to achieve better convergence whenthe added-mass effect is important thus allowing the solving of a wider range of FSIapplications. It is now time to sum up and present in Figure (3) and (2) the resolutionsteps adopted by the ICFD solver when solving FSI problems for the strong and loosecouplings.


Loose FSI coupling

Figure 2: Loose FSI interaction resolution scheme.


Figure 3: Strong FSI interaction resolution scheme.


5.1.5 Watching and controlling the analysis for FSI cases

We have seen in the previous sections that two FSI couplings were available forthe user. In cases of strong FSI coupling and assuming that the maximum level ofoutput information defined in the ICFD CONTROL OUTPUT card has been trig-gered, some information about the Newton loop and the convergence of the coupledstructural and fluid systems can be extracted

T Force: 0.485925E+01 0.332282E-01 -.658934E+00 0.150875E+00PFEM:: It 4PFEM:: FSI Newton Sub-step Dp 2 0.054780387PFEM:: It 4PFEM:: FSI Newton Sub-step Dp 2 0.003279931PFEM:: It 3PFEM:: FSI Newton Sub-step Dp 2 0.000939913T Force: 0.532562E+01 0.359637E-01 -.845310E+00 0.931377E-01PFEM:: It 3PFEM:: FSI Newton Sub-step Dp 3 0.000497924PFEM:: It 3PFEM:: FSI Newton Sub-step Dp 3 0.000088205PFEM:: It 2PFEM:: FSI Newton Sub-step Dp 3 0.000018324T Force: 0.532361E+01 0.359398E-01 -.844209E+00 0.425614E-03Mechanics ConvergedEquilibrium established after 2 iterations

As described in Figure (3), after the first time step, the solver will first solve thefluid equations. The forces computed will then be communicated to the solid me-chanics solver. The three averaged components appear next to T Force along with aresidual value. The solid mechanics solver will then return the node displacementsas boundary conditions for the fluid solver and the procedure will be repeated untilconvergence has been reached ([17]).


5.2 Thermal coupling

5.2.1 Introduction

Heat transfer is a discipline of thermal engineering that concerns the generation, use,conversion, and exchange of thermal energy and heat between physical systems. TheICFD solver offers the possibility to solve and study the behavior of temperatureflow in fluids. Potential applications are numerous and include refrigeration, airconditioning, building heating, motor coolants, defrost or even heat transfer in thehuman body. Furthermore, the ICFD thermal solver is fully coupled with the thermalsolver using a monolithic approach which allows to solve complex problems whereboth heated structures and flows are present and interact together.

5.2.2 The heat equation

The distribution of heat in a given region over time is described by a convection-diffusion equation also called heat equation

∂T

∂t+ uj

∂T

∂xj− α ∂2T

∂xjxj= f in Ω (79)

where α is a positive constant called thermal diffusivity and f is a potential sourceof heat.

This formulation is incomplete if the appropriate set of boundary conditions andinitial conditions is not specified. The user can specify the temperature on theboundaries resulting in Dirichlet conditions

T (~x, t) = T (~x) on Γ (80)

Furthermore, if no boundary condition is applied, the solver will automatically applya Neumann condition

~n(~x).~∇T (x, t) = 0 on Γ (81)

where ~n(~x) is the normalized normal vector at the ~x point.

5.2.3 Spatial discretization

The time derivative will be approximated with a backward differences of second orderas


1.5T n+1 − 2T n + 0.5T n−1

∆t+ un+1

j

∂T n+1

∂xj− α∂

2T n+1

∂xjxj= fn+1 (82)

The Galerkin method will now be applied to Equation (82) in order to find a weakform that will be integrated over the prescribed domain

∫Ω

Ni

(1.5T n+1 − 2T n + 0.5T n−1

)dΩ +

∫Ω

∆tNiuj∂T n+1

∂xjdΩ−

−∆t

∫Ω

Niα∂

∂xj

(∂T n+1

∂xj

)dΩ = ∆t

∫Ω

Nifn+1dΩ,

(83)

Resulting in the matrix form

M

(1.5T n+1 − 2T n + 0.5T n−1

)+ ∆tS(uni )T n+1 −∆tK(α)T n+1 = ∆tF, (84)

As in Section (4.4.2), a similar convection term stabilization process has to be per-formed in order to get to the final system

M(1.5Tn+1 − 2Tn + 0.5Tn−1) + ∆tS(uni )Tn+1 −∆tK(α)Tn+1 − τ(Sy(uni )Tnproj − Su(uni )Tn+1) = 0,(85)

MTnproj −C(uni )Tn = ∆tF, (86)

5.2.4 Coupling with the thermal solver

In Section (5.1.1), the different types of fluid structure couplings were presented.The same vocabulary can be employed to describe the different couplings betweenthe heat equations solved in the fluid by the ICFD solver and the heat equationssolved by the thermal solver in the structure. For the thermal coupling, a monolithicapproach has been adopted. The coupling between the structure and the fluid istherefore very tight and strong at the fluid-structure interface. The resulting fullsystem includes both the structural and the fluid temperature unknowns (See Figure(4)) and is solved using a direct solver which may in some cases be computer-timeconsuming.


Figure 4: Vector of temperature unknowns when the fluid thermal solver and struc-ture thermal solvers are coupled using a monolithic approach.

5.2.5 Watching and controlling the analysis for thermal analysis.

5.3 Summary

Figure (5) offers a simplified view of the interactions and couplings of the mechanical,thermal and ICFD solvers.

Figure 5: Summary of the different possible interactions between solvers


6 Free surface and mutliphase handling

6.1 The level set method

6.1.1 Introduction

A large collection of fluid problems involves moving interfaces. Applications includeair-water dynamics, breaking surface waves and solid bodies penetrating in fluids.In many such applications, the interplay between the interface dynamics and thesurrounding fluid motion is subtle, with factors such as density ratios and tempera-ture jumps across the interface, surface-tension effects, topological connectivity, andboundary conditions playing significant roles in the dynamics. The solver uses a levelset method based on [24], a fast and reliable technique in order to track and correctlyrepresent moving interfaces.

6.1.2 The level set function

Let us first start by introducing the implicit function φ whose zero isocontour, φ = 0represents the interface. The implicit function φ is defined throughout the wholecomputational domain while the isocontour defining the interface is one dimensionlower. As a convention the fluid domain where the Navier-Stokes equations will besolved (see Section (4.1)) is defined by φ > 0 while the vacuum is defined by φ < 0.Additionally, φ will be described as a distance function resulting in the level setfunction

φ(~x) = min(|~x− ~xI |) on Ω+, for all ~xI ∈ ∂Ω, (87)

φ(~x) = 0 on ∂Ω, for all ~xI ∈ ∂Ω, (88)

φ(~x) = −min(|~x− ~xI |) on ∂Ω−, for all ~xI ∈ ∂Ω. (89)

Furthermore, since φ is Euclidean distance:

|~∇φ| = 1, (90)

This is shown in Figure (6) with the classical dam example where the fluid (φ > 0)and the vacuum (φ < 0) are divided by the interface φ = 0. In the case of two phasefluids, the problem is similar except that the Navier-Stokes equations are solved inthe two domains with a smoothing of the density and viscosity values at the interface.


6.1.3 The convection equation

Now, let us suppose that the velocity of each points on the implicit surface is givenas ~V (~x) i.e ~V (~x) is known for every point ~x with φ(~x) = 0. The simplest way to moveall the points on the surface with this velocity would be to move the nodes located onthe interface by their velocity V through the grid (see Figure (6)). This Lagrangianformulation would not be too hard to accomplish ([24]) if the connectivity does notchange and the surface elements are not distorted too much. However, even smallvelocity fields could cause large distortion of the interface elements and the accuracyof the method can deteriorate quickly if a frequent and regular re-meshing of thedomain is not applied. This need for frequent re-meshing implies higher computercosts and less scalability when running with multiple processors.

In order to avoid these problems, the implicit function φ will be used both to representthe interface and to evolve the interface (see Figure (6)). In order to define theevolution of the implicit function φ, the simple convection equation will be used

∂φ

∂t+ ~V .~∇φ = 0, (91)

Equation (91) defines the motion of the interface where φ(~x) = 0. It is an Eulerianformulation of the interface evolution ([24]) since the interface is captured by theimplicit function φ as opposed to being tracked by the displacement of the interfacenodes as was done in the Lagrangian formulation.

6.1.4 Numerical integration

As in Section (4.3), the Galerkin method will be applied to Equation (91) in orderto find a weak form that will be integrated over the prescribed domain∫

Ω

Ni

(1.5φn+1

i − 2φni + 0.5φn−1i

)dΩ +

∫Ω

∆tNiunj

∂φn+1i

∂xjdΩ = 0, (92)

resulting in the matrix form

M

(1.5φn+1

i − 2φni + 0.5φn−1i

)+ ∆tS(uni )φn+1

j = 0, (93)

As in Section (4.4.2), a similar convection term stabilization process has to be per-formed in order to get to the final system


Dam problem : Initial state

ICFD solver methodNot used by the ICFD

solver

Lagrangian formulation : Grid node

displacements

Eulerian formulation : Level set

method

Figure 6: Free surface dam breaking problem : description of the different interfacetracking methods


M

(1.5φn+1

i − 2φni + 0.5φn−1i

)+ ∆tS(uni )φn+1

j − τ(

Sy(uni )ψnj − Su(u

ni )φn+1

j

)= 0,(94)

Mψni −C(uni )φnj = 0, (95)


7 Anistropic/Isotropic Generalized Flow Through

Porous Media

7.1 Generalized Porous Media Model: isotropic media

A generalization of the Navier-Stokes equations (see [23]) that will allow the definitionof sub-domains with different permeability/porosity by means of the *ICFD PARTLSDYNA keyword was developed. The OSS|SUPG stabilizing Finite Element Methodfor the spatial approximation and the second order Fractional Step Method for thetime integration was adopted.

Being ε, κ the porosity and the permeability of the porous media respectively, we candefine:

ε = void volume/total volume, (96)

and being ui the volume averaged velocity field defined in terms of the fluid velocityfield (uif ) as

ui = εuif , (97)

the generalized flow equations of momentum and mass conservation can be stressedas follow (Einstein sum convention is used for i = 1, 2, 3),

∂ui∂xi

= 0, (98)

ρ

ε

[∂ui∂t

+∂

∂xj

(uiujε

)]= −1

ε

∂(pε)

∂xi+µ

ε

(∂2ui∂xj∂xj

)+ ρgi −Di, (99)

where Di are the forces exerted to the fluid by the porous matrix. For the isotropicmodel the porous forces are a function of the matrix porosity and its permeability.For the isotropic model two well-known models are available:

- Model 1: Ergun correlation Di =µuiκ

+1.75ρ|u|√150√κε3/2

ui, (100)

- Model 2: Darcy-Forchheimer Di =µuiκ

+Fερ|u|√

κui. (101)

where F is the Forchhmeimer inertia parameter. Porous forces Di can be treated asan anistropic tensor (see section 7.2).


7.1.1 Model Parameters limits

• If the porous media is highly permeable (very high porosity): ε → 1, κ →∞ ⇒ Di = 0,

Navier-Stokes is recovered ρ

[∂ui∂t

+∂

∂xj(uiuj)

]= − ∂p

∂xi+ µ

∂2ui∂xj∂xj

+ ρgi,

• If the porous media has a very low porosity: ε → 0, κ → 0 ⇒ ui →small, uiuj → even smaller and Di becomes an important term,

Darcy law is recovered∂p

∂xi= −µui

κ+ ρgi

7.2 Anisotropic Generalized Flow Through Porous Media.

The anisotropic version of the Darcy-Forchheimer term in eq. (99) can be written as

Di = µBijuj + Fερ|u|Cijuj, (102)

being

Bij = (Kij)−1 , (103)

Cij = (Kij)−1/2 , (104)

where Kij is the anisotropic permeability tensor, and F is the Forchhmeimer inertiaparameter.

7.3 Heat Transfer in Porous Media: coupled PM/NS heattransfer.

Being T the temperature field and ui the fluid velocity field, the heat transportequation in the porous media can be written as

ρcpeff

∂T

∂t+ ui

∂T

∂xi= keff

∂2ui∂xj∂xj

, (105)

where

cpeff= [ε(ρflcpfl

) + (1− ε)(ρscps)], (106)

keff = [εkfl + (1− ε)ks]. (107)


cpeffis the porous media effective specific heat capacity, cpfl

the fluid specific heatcapacity, cps the solid (or porous matrix) specific heat capacity, ρfl the fluid density,and ρs the solid (or porous matrix) density. keff is the porous media effective thermalconductivity, kfl the fluid thermal conductivity, and ks the solid (or porous matrix)thermal conductivity. For adiabatic porous media problems cpeff

= ρflcpfland keff =

kfl.

7.4 Computing porous parameters using the pressure-velocity(p− u) experimental data

We assume that the pressure-velocity experimental curve was obtained applying apressure difference (or pressure drop, ∆p = pb − pa) on both ends of a porous slab(thickness ∆x) with porous properties κ and ε (the permeability and porosity re-spectively), see Figure (7).

pb

pa

ΔxFigure 7: Experimental Porous Slab

For instance, the P–V experimental data are defined as in Table 4 (see also Fig-ure (8)).So that, we can fit that experimental curve with a quadratic polynomial of the form∆p(u) = αu2

x + βux, being ux the velocity along the x-axis. For the example above,the fitting curve has the following expression:

∆p = 2u2x + 0.75ux (108)

From the governing equations (98), we can write an expression for the dynamicsof this experiment where pressure gradient is in equilibrium with Darcy and Forch-


Table 4: Pressure – Velocity experimental datau[m/sec] ∆p[N/m2]

0 0.02. 9.58. 134.0

12. 297.026. 1371.530. 1822.534. 2337.538. 2916.5

Figure 8: Pressure-Velocity experimental curve.

heimer terms (we also neglected the effect of gravity forces). So that,

∂p

∂xi= −Di = −µ

κui −

Fερ|u|√κ

ui (109)

with i = 1, 2, 3 for general 3D problems.

We assume that for the experiment the flow is along the x direction, so that equa-tion (109) becomes:

dp

dx= −µ

κux −

Fερ√κu2x. (110)

Now, we can simply integrate equation (110) along x direction, inside the porous


slab probe of ∆x thickness obtaining an expression for ∆p across the probe:

∆p = −Di∆x = −µκux∆x−

Fερ√κu2x∆x. (111)

We recall that for this test along x-axis the ∆p (or the pressure drop) decreases as∆x increases, so using the experimental data in equation (108) and the theoreticalmodel in equation (111) we can write (∆p)experimental = (−∆p)theory, so that:

2u2x + 0.75ux =

µ

κux∆x+

Fερ√κu2x∆x (112)

From the last equation (eq. (112)) we obtain the following relationships:

0.75ux =µ

κux∆x (113)

2u2x =

Fερ√κu2x∆x (114)

that is

0.75 =µ

κ∆x (115)

2 =Fερ√κ

∆x (116)

From the first relationship (eq. (113) or eq. (115)) we can solve for the permeabilityκ (please note those relationships are independent of the velocity ux):

κ =µ

0.75∆x (117)

being the fluid viscosity µ and the probe thickness ∆x known from the experiment.From the second relationship (eq. (114) or eq. (116)) we can solve for the Forchheimerfactor F :

F =2√κ

ρε∆x(118)

being the fluid density ρ known from the experiment (κ was already computed usingthe previous relationship). We recall that the porosity parameter ε is also known fromthe experiment and it is just a geometrical quantity (ε = void volume/total volume).In summary: users can provide the P–V curves from experiments and they will beused to compute the permeability κ and the Forchheimer factor F . For this purposes,we will allow the setting of the P–V data using a LOAD_CURVE. So, we will introducein LS-DYNA a new porous model to put in the 4th card of the ICFD_MAT keywordpointing to the load curve in which the experimental data (P–V curve) is defined.


8 Turbulence models

8.1 The RANS model

8.1.1 Introduction

Reynolds-averaged Navier-Stokes (RANS) equations are the most classic approach toturbulence modeling. An time averaged version of the governing equations is solved,which introduces new apparent stresses known as Reynolds stresses. This adds asecond order tensor of unknowns for which various models can provide differentlevels of closure. The incompressible solver provides a k-ε model which is one ofthe most widely used turbulence models in CFD. This model adds two additionalvariables, the turbulent kinetic energy k, and the turbulent dissipation ε and providestwo additional equations that close the model and account for history effects likeconvection and diffusion of turbulent energy. Since RANS models consider a timeaveraged version of the governing equations, such models will be more adequatewhere transient effects are not important and a stationary solution can be reached.

8.1.2 The Reynolds Averaged Equations

When solving Equation (10) and (11), instantaneous quantities are being considered.In order to define the RANS model, the flow will be divided in two parts, a mean(oraverage) component and a fluctuating component. Thus the instantaneous velocityand pressure can be written as,

ui = ui + u′i (119)

p = P + p′. (120)

This technique for decomposing the instantaneous motion is referred to as the Reynoldsdecomposition. Substitution of Equations (119) and (120) in (10) and (11) yields

ρ(∂ui + u′i∂t

+ (uj + u′j)∂(ui + u′i)

∂xj) = −∂(P + p′)

∂xi+ µ

∂2(ui + u′i)

∂xj∂xjin Ω,(121)

∂(ui + u′i)

∂xi= 0 in Ω. (122)

Equations (121) and (122) can now be averaged to yield an equation expressingmomentum conservation for the average motion. By noting that the average of a


derivative is the same as the derivative of the average and that the time average ofthe fluctuating quantity is zero, resulting in the following Reynolds-averaged NavierStokes equations (or RANS equations)

ρ(∂ui∂t

+ (uj)∂(ui)

∂xj) = −∂(P )

∂xi+

∂

∂xj

[µ∂ui∂xj− ρu′iu′j

]in Ω, (123)

∂(ui)

∂xi= 0 in Ω. (124)

As can be seen from Equations (123) and (124), the time averaged equations arevery similar to the instantaneous equations but for the apparition of a new apparentviscous stress term, ρu′iu

′j owing to the fluctuating velocity field generally referred to

as the Reynolds stresses. The aim of the different RANS models will be, under givenhypotheses, to provide additional equations that allow the solving of the system.

8.1.3 The k-ε model

For the moment, the standard k − ε model is implemented, but more models maybe implemented in the future. The k − ε model is two equation model based on theBoussinesq hypothesis that stipulates that the momentum transfer caused by turbu-lent eddies can be modeled with an eddy viscosity allowing to model the Reynoldsstresses as,

ρu′iu′j = 2µtSi,j −

2

3ρkδi,j (125)

µt = ρCµk2

ε(126)

Si,j =1

2

∂ui∂xj− 1

3

∂uk∂xk

δi,j (127)

where µt is the eddy viscosity, Si,j is the mean strain rate, k the mean turbulentkinetic energy and ε the turbulent dissipation. The k − ε model then provides thetwo mandatory transport equations in order to close the model

∂ρk

∂t+∂(ρkui)

∂xi=

∂

∂xj

[(µ+

µtσk

)∂k

∂xj

]+ µtS

2 − ρε, (128)

∂ρε

∂t+∂(ρεui)

∂xi=

∂

∂xj

[(µ+

µtσε

)∂ε

∂xj

]+ C1ε

ε

kµtS

2 − C2ερε2

k, (129)


with the following default constants that can be changed by the user through theICFD CONTROL TURBULENCE card:

C1ε = 1.44 (130)

C2ε = 1.92 (131)

σε = 1.3 (132)

σk = 1.3 (133)

Cµ = 0.09 (134)

(135)

8.2 The LES model

8.2.1 Introduction

As the power of computer increases Large Eddy Simulation (LES) models have be-come a popular technique in order to simulate turbulence. Those models are basedon the assumption that large eddies contain most of the kinetic energy of the flowand depend on the geometry while the smaller ones are considered more universaland independent of the flow’s geometry. Therefore, instead of using time averagedquantities as in RANS models, LES models divide each flow variable into a large-scale (resolved) component and into a subgrid (estimated) component. The resolvedcomponents are still time-dependent allowing the analysis of instantaneous quanti-ties even in flows that are stationary in the average, The filtering is usually meshsize dependent and the model used by the ICFD solver is the classical Smagorinskymodel.

8.2.2 LES equations

The quantities will be divided in a resolvable scale part Ui and in a subgrid scalepart usubi

ui = Ui + usgsi (136)

p = P + psgs (137)

Applying this decomposition to Equation (10) yields after filtering the resultingequation


ρ(∂Ui∂t

+ (Uj)∂(Ui)

∂xj) = −∂P

∂xi+

∂

∂xj

[µ∂Ui∂xj

+ ρτi,j

]in Ω (138)

with the extra stress term due to the sub grid scales (SGS)

τi,j = UiUj − UiUj (139)

Subgrid-scale turbulence models usually employ the Boussinesq hypothesis in orderto express the SGS stress as a turbulent viscosity µsgs

ρ(∂Ui∂t

+ (Uj)∂(Ui)

∂xj) = −∂P

∂xi+

∂

∂xj

[µ∂Ui∂xj

+ µsgs∂Ui∂xj

]in Ω (140)

8.2.3 The Smagorinsky model

The Smagorinsky model describes the turbulent viscosity as

µsgs = ρ(Cs∆)2|S| (141)

where the filter width is usually taken as

∆ = (Volume)13 (142)

and

S =√

2Si,jSi,j (143)

Cs is a constant named Smagorinsky constant. Its default value in the solver is 0.1and can be changed in the ICFD CONTROL TURBULENCE card.

8.3 Wall function

By default, the turbulent viscosity is applied on the whole fluid domain. However,there physically exists a zone in the boundary layer of the fluid very close to thesolid boundary or wall where the fluid is laminar. This may heavily influence onthe calculated friction drag. In order to have a chance to capture this laminar sub-boundary layer, a near wall damping factor proposed by Van Driest ([34]) will beintroduced in the turbulent viscosity term

fν = 1− e−y+

A+ (144)


where A+ is a constant and y+ a non-dimensional wall distance for a wall-boundedflow defined as

y+ =u∗y

ν(145)

where u∗ =√

τwρ

the friction velocity and τw = µ(∂u∂y

) is the shear wall stress.

8.4 Turbulence synthesis method for Large Eddy Simula-tions

A well known synthetic turbulence generator [4] that employs Fourier techniques isthe random flow generation (RFG) method proposed by Smirnov et al. [2]. Devel-oped on the basis of the work of Kraichnan [19], this methodology involves scal-ing and orthogonal transformations where a transient flow field is generated in athree-dimensional domain as a superposition of harmonic functions with randomcoefficients. The method can generate an isotropic divergence-free fluctuating veloc-ity field satisfying the Gaussian’s spectral model as well as an inhomogeneous andanisotropic turbulence flow, provided that an anisotropic velocity correlation tensoris given. Smirnov et al. [2] used their approach to set inlet boundary conditions toLES methods in the simulation of turbulent fluctuations in a ship wake as well asinitial boundary conditions in the simulation of turbulent flow around a ship-hull.Another application successfully tested by the authors was the particle dynamicsmodeling (see [1]).The procedure of generating turbulent condition at inlet walls is:

1. For a turbulent flow field ui(xj, t)i,j=1..3 characterized by its symmetric anistropicvelocity correlation tensor rij = uiuj, look for an orthogonal transformation aijthat diagonalize rij and where the following relationships hold,

amianjrij = δmnc2(n), (146)

aikakj = δij. (147)

As we will see ci coefficients are related to the velocity fluctuations field (u′, v′, w′)in the space generated by the transformation aij.

2. Generate a turbulent velocity field vi(xj, t)i,j=1..3 using a modification of theKraichnan method,

vi(x, t) =

√2

N

N∑n=1

[pni cos(knj xj + ωnt) + qni sin(knj xj + ωnt)], (148)


where

x =x

l, t =

t

τ, knj = knj

c

c(j)

, c =l

τ, (149)

pni = εijmζnj k

nm, qni = εijmξ

nj k

nm (150)

ζni , ξni , ωn ∈ N(0, 1); kni ∈ N(0, 1/2), (151)

where l, τ are the length and time scales of turbulence, εijm is the permutationtensor and N(M,σ) is a normal distribution with mean M and standar devia-tion σ. kni and ωn represent a sample of n wavenumber vectors and frequenciesof the modeled turbulence spectrum

E(k) = 16

(2

π

)1/2

k4 exp(−2k2). (152)

3. Scale and transform vi generated in the previous step to obtain the turbulentflow field ui

wi = civi, (153)

ui = aikwk. (154)

The input of this procedure are the turbulent intensities Iu, Iv, Iw (i.e.√rij = uavg ·

diagIu, Iv, Iw in the transformed principal axes) and the turbulence lenght scale l.These are introduced in the LS-DYNA model using ICFD CONTROL TURB SYNTHESISkeyword in combination to VAD=4 in the card ICFD BOUNDARY PRESCRIBED VEL.

8.5 The Random Flow Generator in Python: How Inlet tur-bulence generator works (basically) in ICFD/LS-DYNA

Import of some python modules and vector normal random generator.

In [57]: import numpy as np

import time

import sys

import math

from pylab import *

import random

from oct2py import Oct2Py, octave

from IPython.core.display import display, Math


Figure 9: Flow around Ahmed body with synthetic turbulent inlet boundary condi-tions.

In [58]: def normalrandomvec(mean,sigma):

x=random.gauss(mean,sigma)

y=random.gauss(mean,sigma)

z=random.gauss(mean,sigma)

return np.array([x,y,z]) # 1x3

8.6 Smirnov’s algorithm to generate inhomogeneousanisotropic turbulence at inflow boundaries in LESruns

i). For a given mean velocity at inlet uavg and given turbulence intensity

I =[Iu = urms

uavg, Iv = vrms

uavg, Iw = wrms

uavg

]In [59]: # u_avg -> mean longitudinal (y axis) wind speed, per node, [m/s]:

u_avg = 23.6

#turbulence intensity Iu != Iv != Iw

Iu = 0.1


Iv = 0.1

Iw = 0.1

urms = u_avg*Iu

vrms = u_avg*Iv

wrms = u_avg*Iw

ii). Given an anisotropic velocity correlation tensor

rij = uiuj

of a turbulent flow field ui(xj, t):find an orthogonal transformation tensor aij that diagonalize rij

amianjrij = c2(n)δmn

amianj = δij

c(n) and aij are function of space. c(n) are the fluctuating velocities in the transformedsystem (principal axes).

In [60]: #anisotropic velocity correlation tensor:

rij = np.array([[urms**2, 0, 0],[0, vrms**2, 0],[0, 0, wrms**2]])

#find an orthogonal transformation tensor aij that would

#diagonalize rij:

aij, c2 = np.linalg.eig(rij)

cn = np.sqrt([aij[0], aij[1], aij[2]]) # 1x3

#L -> turbulence length scale [m]

Lu = Lv = Lw = 0.1

Ls = math.sqrt(Lu**2 + Lv**2 + Lw**2)

#Ls = 0.13

In [61]: #N -> sampling number for each wavenumber kn

N = 1000

# x -> nodal coordinates

x = np.array([[0.0],[0.0],[0.0]]) # 3x1

timev = np.arange(0,3.,0.0001) # 4000,


In [62]: print "begin simulation..."

#modified Kraichnan’s method

uxt = np.zeros([3,timev.size]) # 3x4000

pni = np.zeros([3,1]) # 3x1

qni = np.zeros([3,1]) # 3x1

knjtil = np.zeros([3,1]) # 3x1

#time-scale of turbulence [sec]

tau = Ls/u_avg

timetil = timev/tau # 4000,

xtil = x/Ls # 3x1

c = Ls/tau

un = np.zeros([3,timev.size]) # 3x4000

# initialize seed:

random.seed()

for n in range(0,N):

omegamn = random.gauss(0,1)

knj = normalrandomvec(0,0.5) # 1x3

Zetan = normalrandomvec(0,1) # 1x3

Xin = normalrandomvec(0,1) # 1x3

pni = np.cross(Zetan.transpose(),knj.transpose()) # 1x3

qni = np.cross(Xin.transpose(),knj.transpose()) # 1x3

knjtil[0,0] = knj[0]*c/cn[0]



un[0,:] = un[0,:] +

+ pni[0]*cos(np.inner(knjtil.T,xtil.T) + omegamn*timetil[:]) +

+ qni[0]*sin(np.inner(knjtil.T,xtil.T) + omegamn*timetil[:])

un[1,:] = un[1,:] +



un[2,:] = un[2,:] +




#print(n)

uxt[0,:] = cn[0]*math.sqrt(2./N)*un[0,:]



print "end simulation..."

begin simulation...

end simulation...

In [63]: um = max(uxt[0,:])

vm = max(uxt[1,:])

wm = max(uxt[2,:])

subplot(311)

plot(timev,uxt[0,:]/u_avg,color="red",linewidth=1)

axis([0, 3, -um/u_avg, um/u_avg])

title(’Turbulence Syntheis’)

xlabel(’time [secs]’)

ylabel(’$u_x$’,fontsize=20)

grid(True)

subplot(312)

plot(timev,uxt[1,:]/u_avg,color="blue",linewidth=1)

axis([0, 3, -vm/u_avg, vm/u_avg])


ylabel(’$u_y$’,fontsize=20)

grid(True)

subplot(313)

plot(timev,uxt[2,:]/u_avg,color="green",linewidth=1)

axis([0, 3, -wm/u_avg, wm/u_avg])


ylabel(’$u_z$’, fontsize=20)

grid(True)

# simulated fluctuations

oc = Oct2Py()


octave.addpath(’/shang2_2/rpaz/LS-DYNA/DOCUMENTS/IMPLEMENT/SINTESIS/gen_spectrum.m’)

fff,pf = oc.gen_spectrum(uxt,timev)

# theoretical fluctuations

kmax = 100

k = np.arange(0.1,kmax,0.5) # 400,

#k = f/u_avg

Ek = 16.*math.sqrt(2./math.pi)*(k/u_avg)**4.*(np.exp(-2.*(k/u_avg)**2.))

fig,ax = subplots()

ax.loglog(fff[0,:],pf[0,:],"b",linewidth=1)

ax.loglog(k,Ek,"r",linewidth=1)

ax.axis([1, 1000, 1.e-10, 100])

ax.set_xlabel(’$k$ [1/hz]’,fontsize=20)

ax.set_ylabel(’$E(k)$’, fontsize=20)

ax.set_title(’Spectra’, fontsize=20)

ax.legend(["Simulated Spectrum","Theoretical Spectrum"]);

ax.grid(True)


In [64]: subplot(311)

plot(timev,u_avg+uxt[0,:],color="red",linewidth=1)

axis([0, 3, u_avg-um, u_avg+um])

title(’Turbulence Syntheis’)


ylabel(’$U_x$ [m/s]’,fontsize=12)

grid(True)

subplot(312)

plot(timev,uxt[1,:],color="blue",linewidth=1)

axis([0, 3, -vm, vm])


ylabel(’$U_y$ [m/s]’,fontsize=12)

grid(True)

subplot(313)

plot(timev,uxt[2,:],color="green",linewidth=1)

axis([0, 3, -wm, wm])


ylabel(’$U_z$ [m/s]’, fontsize=12)

grid(True)



9 The Volume Mesher

9.1 Introduction

The ICFD solver uses an automatic volume mesher for the fluid domains. Thisgreatly simplifies the pre-processing stage. For this feature a good quality body-fitted surface mesh has to be provided. For FSI simulations, the solver uses anALE approach for mesh movement. In the cases where FSI simulations result inlarge displacements the solver can automatically re-mesh to keep an acceptable meshquality. This section will focus on giving the basic steps of how the volume mesh isbuilt as well as describe some local refinement and re-meshing tools. The meshingtechnique adopted by the solver is inspired by ([18]).

9.2 The Delaunay criteria

Before describing the steps leading to the construction of the volume mesh, let usintroduce the so-called Delaunay criterion as its application will be fundamental inthe construction of the volume mesh. In mathematics and computational geometry,a Delaunay triangulation for a set of nodes P in a plane is a triangulation DT(P)such that no node is inside the circumcircle of any triangle DT(P). For instance,in Figure (10), the Delaunay criterion is violated since the node P4 is inside thecircumcircle of the triangle composed of P1, P2 and P3.

P1

P2

P3

P4

Figure 10: Violation of the Delaunay criterion, the point P4 is inside the circumcircleof the triangle P1P2P3

9.3 Initial Volume mesh building steps

• Step one : The solver reads the surface nodes and elements given by theuser. The list of surfaces have to be non-overlapping and should not leave


any gaps or open spaces between the surface boundaries. The nodes on theboundary of two neighbor surfaces have to be uniquely defined (no duplicatenodes) and should match exactly on the interface. Each surface element hasto be orientable i.e a normal vector defining the interior/exterior faces of thesurface element can be associated to it. If one of those conditions is not met,the solver will return an error.

• Step two : Using the initial surface nodes, the solver will then join the surfacenodes in order to build an initial volume mesh. The tetrahedras or trianglesin 2D thus created all need to respect the Delaunay criteria.

• Step three : The solver will progressively add nodes to the volume mesh.Every time a new node is added, its ”host” tetrahedra or triangle will bedivided in smaller tetrahedras or triangles respecting the Delaunay criteria.This procedure will be repeated until the volume elements have reached theirdesired size based on a linear interpolation of the surface sizes that define thevolume enclosure. Figure (11) shows an example of the different steps leadingto the construction of the initial volume mesh.

9.4 Local refinement tools

9.4.1 Imposing the mesh size on a volume domain

During the geometry set up, the user can define surfaces that will be used by themesher to specify a local mesh size inside the volume. Two cards and methods areavailable:

• For the MESH SIZE card, a surface with the specified element size must bedefined in the input deck and will be read along with the other initial surfaces(Step one). This surface must be entirely defined in the volume mesh butdoes not need to be necessarily closed. The nodes of this specific surfacewill directly be used when building the initial volume mesh (Step two). Alinear interpolation will then be used to define the element sizes between thisspecifically added surface and the classic surfaces defining the volume enclosure.

• For the MESH SIZE SHAPE card, a local element size can be defined by theuser in specific zones corresponding to given geometrical shapes (box, sphere,


Step 1

Step 2

Step 3

Figure 11: Initial Volume mesh building steps


cylinder). This element size will be used as reference in the specified zone whenprogressively adding nodes and building the final volume mesh (Step three).This zone does not necessarily need to be entirely defined in the volume mesh.However, in cases where a boundary surface mesh is defined in this zone, itis advised to use a similar element size for the shape zone and the boundarysurface mesh in order to keep a good quality mesh.

Figure (12) shows an example of the two methods and the resulting volume meshusing the same rectangular shape and element size in both cases.

9.4.2 The boundary layer mesh

It is also possible to specify several anisotropic elements to be added to the boundarylayer in order to better represent close-to-the-wall effects. The solver will extrude thespecified surface mesh in the normal direction of the surface elements by a certaindistance proportional to their size. The desired number of boundary layer elementswill then be added to this thickness by always dividing the element closest to thesurface thus resulting in a boundary layer mesh that gets finer when approaching thewall.

9.5 Volume mesh remeshing criteria

9.5.1 Mesh deformation criteria

For FSI simulations, the solver uses an ALE approach for mesh movement whichmeans that large deformations of the fluid mesh can occur. By default, the solver,only rebuilds the mesh if elements get inverted. Inversion of elements usually occurin rotational problems such as wind turbine analyzes. However, it is also possiblethrough the ICFD CONTROL ADAPT SIZE card to trigger a re-meshing of ele-ments that have been distorted by the mesh movement algorithm and that no longerrespect the initial mesh size. This frequently occurs if problems involving bodiesmoving in translation.

9.5.2 Using adaptivity

Using the keyword ICFD CONTROL ADAPT, the user can also trigger an automaticremeshing where the solver will use an a-posteriori error estimator to compute a newmesh size bounded by the user to satisfy a maximum perceptual error. The errorestimator and adaptive remeshing procedure is based on the work by [36]. Therelative percentage error will be defined as


Step 1

Step 2

Step 3

Using the MESH SIZE SHAPE card

Step 2

Step 1

Step 3

Using the MESH SIZE card

Figure 12: Locally imposing the mesh size : comparison between the MESH SIZEcard and the MESH SIZE BOX card


η =||e||||φ||

(155)

where φ can be pressure, velocity, temperature or even the level set function and ecan be written as

e = φ− φ (156)

where e is the difference between the exact solution φ and the approximate finiteelement solution φ. In order to estimate the exact solution, an L2 projection methodwill be employed.For every nodal value, it is therefore possible to associate an error value. As a result,if too many nodes reach the critical value set by the user, the solver will apply aremeshing and refine in the zones where the error was high.

9.6 Watching the mesh quality

9.6.1 Mesh quality output file

Once the initial volume mesh has been built, it is often important to check itsquality. Assuming that the minimum file output level has been defined in theICFD CONTROL OUTPUT card, an mstats file will be generated. In this file,the user can check the total number of nodes and elements generated, their overallquality as well as the amount of distorted elements. Each time a remeshing is made,a new mstats file will be generated.

Nodes: 310Tris : 572Total Area : 104.00000000******* Element Quality *******Angles:Min. Angle = 21.81625125Max. Angle = 129.73094855Avg. Min. Angle = 46.46269865Avg. Max. Angle = 77.06048470minang< 5 = 0 0.000000%George Quality(L*S/V) (Q = big => sliver, Q = 1 => nice Tet):Min. Q = 0.69308732Max. Q = 1.80546082Avg. Q = 0.87247972Q>10= 0 0.000000%


9.6.2 Terminal information

During the analysis, several additional information is also displayed on the terminalwindow in cases of moving mesh or adaptivity if the maximum level of output infor-mation has been triggered in the ICFD CONTROL OUTPUT card. The followingfigure shows the maximum outputs that can appear during the run. Most of thisinformation is mainly for debugging purposes but it can give some hints on the meshquality to users.

Minimum Jacobian(x): 0.908503476738054Minimum Jacobian(u): 0.989910731457378Maximum Derivative: 42.9609250334868Adapt ratios(rat(1), rat(2), reno): 0.102040816326531 0.176038288090357 0.000000000000000E+000Adapt ratios(rat(1), rat(2), reno): 0.000000000000000E+000 0.000000000000000E+000 0Adapt ratios(E/tlm, E1/tlm, Total): 0.533582507931422 5.374310168412143E-002 0.587325609615544

The Minimum Jacobian(x) gives some indication on the mesh distortion comparedto its initial position. The Minimum Jacobian(u) should always stay positive as anegative value means an inversion of an element. The Maximum Derivative gives alsosome indication on the mesh distortion, a very large value could mean a very deformedelement that may lead to a non-convergence. The following two Adapt ratios(rat(1),rat(2), reno) lines only appear in cases where the ICFD CONTROL ADAPT SIZEor ICFD CONTROL ADAPT cards are being used and give some information onthe nodal and element deformation respectfully. rat(1) symbolizes the percentage ofnodes that need remeshing, rat(2) the fluid volume percentage of these nodes anda positive value of reno indicated a heavily distorted element. If one of those threevalues reach a certain criteria, a remeshing will be applied. The Adapt ratios(E/tlm,E1/tlm, Total) only appears in cases where the ICFD CONTROL ADAPT card isused. It gives some information on the volume percentage of the fluid domain thatneeds to be refined (E/tlm), the volume percentage of the fluid domain that can becoarsened (E1/tlm) as well as the sum of the two values Total. If the Total valuereaches a certain criteria, a remeshing will be applied.


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[5] P. Causin, J.-F. Gerbeau, and F. Nobile, Added-mass effect in the designof partitioned algorithms for fluid-structure problems., Comp. Meth. Appl. Mech.Engrg.,, 194 (2005), pp. 4506–4527.

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A Consistent units for ICFD

USI Equivalence ([kg]α ∗ [m]β ∗ [s]γ) ex 1 ex 2

Mass kg [kg]α [m]β [s]γ g g

Length m mm mm

Time s s ms

Energy J 1 2 -2 1e−9 1e−3

Force N 1 1 -2 1e−6 1

Stress/Pressure Pa 1 -1 -2 1 1e6

Velocity ms

0 1 -1 1e−3 1

Density kgm3 1 -3 0 1e6 1e6

Dynamic Viscosity Pa s 1 -1 -1 1 1e3

Thermal Diffu. m2

s0 2 -1 1e−6 1e−3

Heat capacity JkgK

0 2 -2 1e−6 1

Thermal Cond. Jms

1 1 -3 1e−6 1e3

Table 5: Consistent units for the ICFD solver


Documents

Incompressible uid solver in LS-DYNAftp.lstc.com/anonymous/outgoing/inaki/docs/pdf_icfd/ICFD...Test Case Name Incompressible uid solver in LS-DYNA Test Case ID ICFD-THE-1.1 Test Case