Low-complexity stochastic modeling of spatially-evolving flows

Wei Ran, Armin Zare, M. J. Philipp Hack, and Mihailo R. Jovanovic

Abstract— Low-complexity approximations of the Navier-Stokes (NS) equations are commonly used for analysis andcontrol of turbulent flows. In particular, stochastically-forcedlinearized models have been successfully employed to capturestructural and statistical features observed in experiments andhigh-fidelity simulations. In this work, we utilize stochastically-forced linearized NS equations and the parabolized stabilityequations to study the dynamics of flow fluctuations in tran-sitional boundary layers. The parabolized model can be usedto efficiently propagate statistics of stochastic disturbances intostatistics of velocity fluctuations. Our study provides insight intothe interaction of the slowly-varying base flow with streamwisestreaks and Tollmien-Schlichting waves. It also offers a sys-tematic, computationally efficient framework for quantifyingthe influence of stochastic excitation sources (e.g., free-streamturbulence and surface roughness) on velocity fluctuations inweakly non-parallel flows.

Index Terms— Boundary layers, distributed systems, energyamplification, parabolized stability equations, stochasticallyforced Navier-Stokes equations.

I. INTRODUCTION

The analysis, optimization, and control of dynamical mod-els that are based on the Navier-Stokes (NS) equations isoften hindered by their complexity and large number of de-grees of freedom. While the existence of coherent structuresin wall-bounded shear flows [1] has inspired the developmentof reduced-order models using data-driven techniques, theimportant features of such models can be crucially alteredby control actuation and sensing. This gives rise to nontrivialchallenges for model-based control design [2].

In contrast, linearization of the NS equations aroundmean-velocity is well-suited for analysis and synthesis usingtools of modern robust control. Linearized models subjectto stochastic excitation have been employed to replicatestructural and statistical features of transitional [3]–[5] andturbulent [6]–[8] wall-bounded shear flows. In these modelsthe nonlinear terms in the NS equations are replaced bystochastic forcing. Most studies have focused on parallel flowconfigurations in which translational invariance allows for thedecoupling of the dynamical equations across streamwise andspanwise wavenumbers. This offers significant computationaladvantages for analysis, optimization, and control.

Financial support from the National Science Foundation under AwardCMMI 1363266 and the Air Force Office of Scientific Research under AwardFA9550-16-1-0009 is gratefully acknowledged.

Wei Ran, Armin Zare, and Mihailo R. Jovanovic are with the Departmentof Electrical Engineering, University of Southern California, Los Angeles,CA 90089. M. J. Philipp Hack is with the Center for Turbulence Re-search, Stanford University, Stanford, CA 94305. E-mails: wran@usc.edu,armin.zare@usc.edu, philipp.hack@stanford.edu, mihailo@usc.edu.

In the flat-plate boundary layer streamwise and normalinhomogeneity leads to the temporal eigenvalue problem forPDEs with two spatial variables. This problem is compu-tationally more difficult to solve than for parallel flows.Previously, tools from sparse linear algebra and iterativeschemes have been employed to analyze the spectra of thegoverning equations and provide insight into the dynamicsof transitional flows [9]–[11]. Efforts have also been madeto conduct non-modal analysis of spatially-evolving flowsincluding transient growth [10], [12] and resolvent [13]analyses. In spite of these successes many challenges remain.

The Parabolized Stability Equations (PSE) result fromthe removal of elliptic components from the NS equations.The PSE respect inhomogeneity in the streamwise directionbut do not propagate information upstream. This makesthem computationally more efficient than conventional flowsimulations based on the NS equations [14]. In particular,the resulting set of equations are convenient for marching inthe downstream direction [15]–[17]. They are thus routinelyused to compute the spatial evolution of instability modes ina wide range of engineering problems [17].

Despite their popularity, parabolized equations have beenutilized in a rather narrow context. We revisit the modelingof spatially-evolving boundary layer flow by examiningthe utility of such models in assessing the receptivity andquantifying the sensitivity to different types of flow distur-bances. Our simulation-free approach lays the groundworkfor a systematic, computationally efficient framework forquantifying the influence of stochastic excitation sources onvelocity fluctuations in weakly non-parallel flows.

The paper is organized as follows. In Section II, wedescribe stochastically-forced linearization of the NS equa-tions around Blasius boundary layer flow. In Section III, westudy the receptivity to stochastic excitations of the velocityfluctuations around locally-parallel and spatially evolvingBlasius profiles. In Section IV, we adopt stochastic forcingto model the effect of excitations in the PSE. Finally, weprovide concluding thoughts in Section V.

II. BACKGROUND

In this section, we present the equations that govern thedynamics of flow fluctuations in incompressible flows ofNewtonian fluids and characterize the structural constraintsthat are imposed on the second-order statistics by the lin-earized dynamics.

In a flat-plate boundary layer, with geometry shown inFig. 1, the dynamics linearized around the Blasius boundary

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Fig. 1: Geometry of a transitional boundary layer flow subjectto stochastic forcing.

layer profile u = [U(x, y) V (x, y) 0 ]T are

vt = − (∇ · u)v − (∇ · v) u − ∇p +1

Re0∆v + f ,

0 = ∇ · v,(1)

where v = [ v1 v2 v3 ]T is the vector of velocity fluc-tuations, p denotes pressure fluctuations, and v1, v2, andv3 represent components of the fluctuating velocity fieldin the streamwise (x), wall-normal (y), and spanwise (z)directions, respectively. The Reynolds number is defined asRe0 = U∞δ0/ν, where δ0 =

√νx0/U∞ is the Blasius

length scale at the inflow, U∞ is the free-stream velocity,and ν is the kinematic viscosity. Spatial coordinates are non-dimensionalized by δ0, velocities by U∞, time by δ0/U∞,and pressure by ρU2

∞, where ρ is the fluid density. Thepresence of the additive zero-mean stochastic body forcing fcan be justified in different ways [3]–[5]. For our purposes,we wish to compensate the role of the neglected nonlinearinteractions by introducing stochastic sources of excitationthat perturb the otherwise linearly developing velocity field.

III. STOCHASTICALLY-FORCED LINEARIZED NSEQUATIONS

In this section, we first examine the dynamics of flowfluctuations in the stochastically-forced Blasius boundarylayer under a locally-parallel base flow assumption. Thisassumption entails linearization around the Blasius profileevaluated at a fixed streamwise location x0. Since the result-ing base flow only depends on the wall-normal coordinate y,this also allows for the parameterization of the correspondingevolution model over horizontal wavenumbers. This reducesthe computational complexity of studying the amplificationof streamwise streaks and Tollmien-Schlichting (TS) waves,which are of importance in the laminar-turbulent transitionof boundary layer flows. We then repeat the same exercisefor the NS equations linearized around a base flow profilewhich is a function of both streamwise and wall-normalcoordinates. In fluids literature this approach is called globalanalysis and it is typically computationally challenging. Wehave ensured grid convergence by doubling the number ofpoints used to discretize the differential operators in thestreamwise and wall-normal coordinates.

A. Parallel Blasius boundary layer flow subject to free-stream turbulence

We perform an input-output analysis to quantify the energyamplification of velocity fluctuations subject to an exogenoussource of excitation that represents free-stream turbulence.This excitation is modeled as white-in-time stochastic forcinginto the linearized NS equations around the parallel Blasiusbase flow profile, i.e., the Blasius profile evaluated at onestreamwise location x0. This choice is motivated by previousstudies which show that transient growth exhibits similartrends for parallel and non-parallel boundary layer flows [18],[19].

Under the assumption of a parallel base flow, translationalinvariance allows us to apply Fourier transform in the plate-parallel directions. This brings the state-space representationof the linearized NS equations to the form

ψ(k, t) = A(k)ψ(k, t) + B(k) f(k, t),v(k, t) = C(k)ψ(k, t).

(2)

Here, ψ = [ vT2 ηT ]T ∈ C2Ny is the state, which containsthe wall-normal velocity v2 and vorticity η , and v ∈ C3Ny

with Ny being the number of collocation points in the finitedimensional approximation of the differential operators inthe wall-normal direction. Equations (2) are parameterizedby the spatial wavenumber pair k = (kx, kz); see [5] forthe expressions of A, B, and C. We consider no-slip andno-penetration boundary conditions.

In statistical steady-state, the covariance matrix

Φ(k) = limt→∞

E (v(k, t)v∗(k, t))

of the velocity fluctuation vector, and the covariance matrix

X(k) = limt→∞

E (ψ(k, t)ψ∗(k, t)) ,

of the state in (2), are related as follows:

Φ = C X C∗.

Here, ∗ denotes complex-conjugate-transpose and E is theexpectation operator. Matrix Φ contains information aboutall second-order statistics of the fluctuating velocity field.

For a stable dynamical generator A, the steady-state co-variance of the state in (2) subject to zero-mean and white-in-time stochastic forcing with covariance Ω = Ω∗ 0, i.e.,

E (f(t1) f∗(t2)) = Ω δ(t1 − t2),

is determined by the solution to the Lyapunov equation,

AX + X A∗ = −B ΩB∗. (3)

The energy amplification of the stochastically-forced flowcan be computed using the solution to the Lyapunov equa-tion (3) as:

E = trace (C X C∗) . (4)

The receptivity to external forcing that enters in variouswall-normal locations can be evaluated by computing theenergy spectrum of the velocity fluctuations. To specify thewall-normal region in which the forcing enters, we define

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y

f(y)

Fig. 2: The shape of the filter function f(y); y1 = 5, y2 = 10.

f := f(y)fs where fs represents the solenoidal forcingarising from free-stream turbulence and f(y) is the smoothfilter function defined as

f(y) :=1

π(atan(y − y1) − atan(y − y2)) . (5)

Here, y1 and y2 determine the shape of f(y); Fig. 2 showsf(y) when y1 = 5 and y2 = 10.

We compute the energy spectrum of stochastically excitedparallel Blasius boundary layer flow with Re0 = 400 (theBlasius length scale is δ0 = 1). Here, we consider a wall-normal region with Ly = 25. A solenoidal white-in-timeexcitation fs is first introduced to the region in the immediatevicinity of the wall by choosing y1 = 0 and y2 = 5 in Eq. (5).Figure 3a shows that the energy of velocity fluctuations ismost amplified at low streamwise wavenumbers (kx ≈ 0)with a global peak at kz ≈ 0.42. Clearly, the energy spectrumis dominated by streamwise elongated flow structures, withTS waves observed at kx ≈ 0.35. As the forcing regionmoves away from the wall the amplification of streamwiseelongated structures persists while the amplification of theTS waves weakens; see Fig. 3b. It is also observed that asthe region of excitation moves away from the wall, energyamplification becomes weaker and the peak of the energyspectrum shifts to lower values of kz . These observationsare in agreement with the global analysis of boundary layerflow presented next.

B. Global analysis of stochastically-forced linearized NSequations

We consider the linearized NS equations around aspatially-evolving Blasius boundary layer profile and intro-duce forcing at various wall-normal locations. To capture thespatially evolving nature of the boundary layer, we employfinite dimensional spatial discretization in both streamwiseand wall-normal directions. The linearized NS generatoris globally stable for the particular Reynolds number andspanwise wavenumbers we consider in this study. Thus, thesteady-state covariance of the fluctuating velocity field canbe obtained from the solution to the Lyapunov equation (3).

kx

kz(a)

kx

kz(b)

Fig. 3: Plots of log10(E(k)) in the Blasius boundary layerflow with Re0 = 400 subject to white-in-time stochasticexcitation entering in various wall-normal regions; (a) y1 =0, y2 = 5 (b) y1 = 5, y2 = 10 in Eq. (5).

The computational region is a rectangular box with Lx ×Ly = 900 × 25 and the initial Reynolds number is Re0 =400. We consider the linearized dynamics given in Eqs. (2)with state ψ = [ vT2 ηT ]T ∈ C2NxNy , k = kz , and differen-tial operators that are now spatially discretized in both wall-normal and streamwise directions. Here, Nx and Ny denotethe number of collocation points in the streamwise and wall-normal directions, respectively. We enforce homogenousDirichlet boundary conditions on η and Dirichlet/Neumannboundary conditions on v2 and introduce Sponge layers in thestreamwise direction to mitigate the influence of boundaryconditions on the fluctuation dynamics [20], [21]. Similarto Section III-A, we assume that white-in-time stochasticforcing is filtered by the function f(y) in (5).

Our computational experiments show that the energyamplification increases as the region of influence for theexternal forcing approaches the wall; for kz = 0.4, theenergy amplification (cf. Eq. (4)) for perturbations that enterin the vicinity of the wall (y1 = 0 and y2 = 5 inEq. (5)) and away from the wall (y1 = 5 and y2 = 10)is 3.9× 106 and 4.0× 105, respectively. Figures 4a and 4bshow the spatial structure of the streamwise component ofthe principal response when white-in-time stochastic forcingenters in the vicinity of the wall and away from the wall,respectively. The streamwise growth of the streaks can beobserved. Figures 4c and 4d display the cross-section ofthese streamwise elongated structures at z = 0. As Figs. 4eand 4f demonstrate, these streaky structures are sandwichedbetween counter-rotating vortical motions in the cross-streamplane; and they contain alternating regions of fast- and slow-moving fluid that are slightly inclined to the wall.

IV. STOCHASTICALLY-FORCED LINEAR PARABOLIZEDSTABILITY EQUATIONS

While it is customary to use the parallel-flow approxi-mation to study the stability of boundary layer flows tosmall amplitude perturbations, this approximation does notaccurately capture the effect of the spatially-evolving baseflow on the stability of the boundary layer. This issue can

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y

zx

(a) streamwise velocity

y

zx

(b) streamwise velocity

y

x

(c) z = 0

y

x

(d) z = 0

y

z

(e) x = 1150

y

z

(f) x = 1150

Fig. 4: Principal modes with kz = 0.4, resulting fromexitation in the vicinity (a,c,e) (y1 = 0 and y2 = 5 in (5)) andaway from the wall (b,d,f) (y1 = 5 and y2 = 10 in (5)). (a,b)Streamwise velocity component where red and blue colorsdenote regions of high and low velocity; (c,d) streamwisevelocity at z = 0; (e,f) y-z slice of streamwise velocity (colorplots) and vorticity (contour lines) at x = 1150.

be addressed by examining the spatial growth of specificwave structures. Furthermore, in the absence of body forcingand neutrally stable modes, linearized models that do notaccount for the spatial evolution of the base flow predicteither asymptotic decay or unbounded growth of fluctuations.On the other hand, the global analysis and direct numericalsimulation of spatially-evolving flows may be prohibitivelyexpensive for analysis and control purposes.

To refine predictions of parallel flow analysis, we utilizethe Parabolized Stability Equations (PSE) to study the dy-namics of flow fluctuations in flat-plate boundary layers.These equations are obtained by removing elliptic compo-nents from the NS equations and they can be easily advanceddownstream via a marching procedure. This approach is sig-nificantly more efficient than conventional flow simulationsbased on the NS equations. In contrast to standard PSE-basedanalysis, we introduce a stochastic forcing term and show

that linear PSE can be used to march covariance matricesdownstream in a computationally efficient manner.

We next provide a brief overview of the stochastically-forced linear PSE. Additional details regarding the PSE canbe found in [15], [16].

A. Linear parabolized stability equations

In weakly non-parallel flows, e.g. the pre-transitionalboundary layer, flow fluctuations can be separated into slowlyand rapidly varying components [15]. This is achieved byconsidering the following decomposition for the fluctuationfield q = [ v1 v2 v3 p ]T in (1),

q(x, y, z, t) = q(x, y)χ(x, z, t) + complex conj.,

χ(x, z, t) = exp (i (θ(x) + kz z − ω t)) ,

θ(x) =

∫ x

x0

α(ξ) dξ,

where q(x, y) and χ(x, z, t) are the shape and phase func-tions, kz and ω are the spanwise wavenumber and temporalfrequency, and α(x) is the streamwise varying generalizationof the wavenumber [15]. The ambiguity arising from thestreamwise variation of both q and α is resolved by imposingthe condition

∫Ωy

q∗qx dy = 0 [15]. The PSE approximationassumes the streamwise variation of q and α as sufficientlysmall to neglect qxx, αxx, αxqx, and their higher orderderivates with respect to x. The stochastically-forced linearPSE thus take the form

Lq + M qx + αxN q = f , (6)

where expressions for operator-valued matrices L, M , andN can be found in [16]. We discretize these operators usinga pseudospectral scheme with Ny Chebyshev collocationpoints in the wall-normal direction [22].

In what follows, we use the PSE to propagate the spatially-evolving state covariance,

X(x) = E (q(x)q∗(x)) ,

via the Lyapunov equation

Xk+1 = Ak+1Xk A∗k+1 + B Ωk+1B

∗, (7)

where E is the expectation operator, k identifies the stream-wise location, and Ωk represents the covariance of thewhite stochastic disturbance f . The dynamical matrix Aand the input matrix B result from a rearrangement ofthe stochastically-forced linear PSE (6) with a constantstreamwise wavenumber α,

qx = (−M−1L)︸ ︷︷ ︸A

q + (−M−1)︸ ︷︷ ︸B

f .

These equations provide a good approximation of pertur-bations with slowly-varying streamwise wavenumbers [23].The streamwise dependence of our equations follows fromthe dependence of the state q, and matrices L and M on thestreamwise location xk. Propagation of the state covarianceXk using Eq. (7) offers significant computational advantage

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|v1| m

ax

x

Fig. 5: Peak of the amplitude of TS waves at ω = 0.0344 ina flat-plate boundary layer flow; nonlinear PSE (solid), linearPSE without forcing (dashed), linear PSE with white forcing(4), and linear PSE with white, but x-dependent forcingdesigned using the procedure explained in Section IV-B (#).

over computation of the covariance from the ensemble aver-age of many stochastic simulations.

We next consider the streamwise evolution of a two-dimensional TS wave and provide a comparison of the resultsobtained using linear PSE with and without stochastic forcingand nonlinear PSE. In this context, the results of nonlinearPSE, which we consider as the ground-truth, closely matchthe results of direct numerical simulations [15].

B. Streamwise evolution of a two-dimensional TS wave

We study the streamwise evolution of a two-dimensionalTS wave (kz = 0) with an initial amplitude of 2.5 × 10−3

and temporal frequency ω = 0.0344. All computations areinitialized at Re0 = 400 with the streamwise wavenumberα and shape function corresponding to the considered TS-mode. This mode is identified by the eigenvalue with thelargest imaginary part in the spatial eigenvalue problem. Thisproblem studies the stability of the linearized NS equationsfor a particular Reynolds number, spanwise wavenumber, andtemporal frequency.

The computational domain is Lx × Ly = 2400 × 60with homogenous Dirichlet boundary conditions applied inthe wall-normal direction. We conduct 400 simulations ofstochastically-forced linear PSE (6), with different realiza-tions of white stochastic forcing. As in Section III-A, wefilter the forcing using the function f(y) in (5) with y1 = 0and y2 = 10. The velocity profiles and covariances thatresult from the ensemble average of these simulations arecompared with the results of linear and nonlinear PSE withno stochastic forcing under the same parameter space andinitial conditions.

Figure 5 shows the peak amplitude of the streamwise ve-locity component of the TS wave. Relative to nonlinear PSE,linear PSE with and without white stochastic forcing under-estimates the streamwise location of the peak. Figure 6 shows

y

y

(a) Xv1v1 , nonlinear PSE

y

(b) Xv1v1

y

y

(c) Xv1v2 , nonlinear PSE

y

(d) Xv1v2

Fig. 6: Velocity covariance matrices at x = 2800 resultingfrom (a,c) simulation of nonlinear PSE, and (b,d) frompropagation of Eq. (7) in Blasius boundary layer flow withTS mode initialization. (a,b) The streamwise correlationmatrix Xv1v1 , and (c,d) the streamwise/wall-normal crosscorrelation matrix Xv1v2 .

covariance matrices of the streamwise and streamwise/wall-normal velocity components at the outflow (x = 2800).These are obtained using simulations of nonlinear PSE andpropagation of Eq. (7). We see that the outflow velocitycovariances resulting from linear PSE with white forcingcapture the essential trends observed in nonlinear PSE.

Our computations show that it is not feasible to exactlymatch velocity correlations and growth trends with an x-independent white stochastic excitation of the linear PSE.This necessitates the use of x-dependent or colored forcing.As we show next, a forcing field which is streamwise depen-dent but uncorrelated in the wall-normal direction enables usto better predict the location of the peak amplitude in Fig. 5.

In order to obtain this forcing, we first use Xk and Xk+1

resulting from nonlinear PSE to compute the covarianceZk := B ΩkB

∗. This covariance corresponds to a spatiallycorrelated process in the streamwise direction and is thusx-dependent and not necessarily positive semi-definite. Wethen project these forcing correlations onto the positive-definite cone to obtain uncorrelated noise in the wall-normaldirection. The result of incorporating this white but x-dependent forcing is also shown in Fig. 5. Clearly, the peakamplitude resulting from this model matches the curve fromnonlinear PSE simulations.

To further evaluate the performance of this model, weexamine the error in matching the full state covariance matrix

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x

(a)y

|v1|(b)

Fig. 7: (a) Relative error in matching the amplitude of stream-wise velocity (solid) and in matching the state covarianceX(x) (dashed) using the linear PSE with x-dependent whiteforcing in the spatial evolution of a 2D TS wave withω = 0.0344. (b) The amplitude of the streamwise componentof a 2D TS waves at x = 1528 from nonlinear PSE (solid)and stochastically forced linear PSE (#).

X and the amplitude of the streamwise velocity profile |v1|as a function of the streamwise location x; see Fig. 7a.This figure shows that the streamwise velocity amplitudeand covariance matrices are reasonably recovered. Figure 7bshows the amplitude of the streamwise velocity |v1| at thelocation with highest error (x = 1528). The profiles resultingfrom linear PSE with x-dependent white forcing perfectlymatch the results of nonlinear PSE.

V. CONCLUDING REMARKS

In the present study, we have utilized stochastically-forcedlinearized NS equations and stochastically-forced linear PSEto study the dynamics of flow fluctuations in the Blasiusboundary layer. In particular, we have examined the recep-tivity of the spatially-evolving Blasius boundary layer flowto free-stream disturbances which we model as stochasticexcitations that enter at specific wall-normal locations. Wealso incorporate stochastic forcing into the linear PSE tostudy the streamwise evolution of TS waves. While whitestochastic excitation is not able to improve predictionsrelative to the conventional linear PSE without stochasticforcing, we achieve better predictions of transient peaksusing white, but x-dependent forcing. The predictive powerof our approach can be further improved by utilizing recentlydeveloped theoretical framework for identifying the spatio-temporal spectrum of stochastic excitation sources [8], [24].The problem of employing such a framework in order torecover partially-observed statistical signatures of spatially-evolving flows via low-complexity stochastic models will bestudied in our future work.

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