On the Divergence-Free Condition of Velocity and

Vorticity in Velocity-Vorticity Formulation of

Incompressible Navier-Stokes Equation

Swagata Bhaumik∗ and Tapan K. Sengupta†

Indian Institute of Technology Kanpur, Kanpur 208 016, INDIA

Evidences gathered from the numerical simulations using derived and primitive variableformulations point unambiguously to the former yielding more accurate numerical results.This is attributed to the fact that vorticity based formulations are able to better resolvefiner scales, as vorticity is defined as the curl of velocity vector. But in the context ofvelocity-vorticity formulation, the point of contention is the satisfaction of the solenoidal-ity condition by both velocity (for 2D as well as 3D flows) and vorticity vectors (for 3D flowsonly). Nonsolenoidality of velocity vector might indicate the presence of distributed sourceand sink throughout the flow-field which might excite the flow in an unphysical mannerresulting in an entirely different simulated flow-field. This might be more severe for phys-ically unstable flows exhibiting spurious instabilities. On the other hand, nonsolenoidalityof vorticity field for three-dimensional flows might imply a presence of distributed vorticescausing unphysical rotational effects. Hence, in order to develop a very accurate methodbased on derived variable formulation for simulating transitional flows, it is essential to as-sess the effects of these nonsolenoidality errors. Here, an attempt has been made to studythis in detail. We have carried out numerical investigations for internal flows at supercrit-ical Reynolds numbers and external flow past a semi-infinite flat plate which is explicitlyexcited. We have shown for 2D as well as 3D flows that solenoidality error in velocity isnot as critical in destabilizing the flow. For simulating 3D lid driven cavity flow at bothsub-critical and super-critical Reynolds numbers, we have used two different variants ofvorticity transport equation and have shown that though at sub-critical Reynolds numberboth variants produce identical results, but at super-critical Reynolds number the variantwhich conserves the solenoidality of vorticity yields better time-accurate results.

Nomenclature

Symbols

∆x , ∆y , ∆z = Grid spacing along x-, y- and z-axis, respectivelyδ∗ = Displacement thicknessω = Component of vorticity vector normal to (x, y)-plane (simply referred to as vorticity for 2D case)

Ω , ~Ω = Vorticity vectorωx, ωy, ωz = x-, y- and z-components of vorticity, respectivelyψ = Stream function (component of vector potential normal to (x, y)-plane)

Ψ , ~Ψ = Vector potentialk = Wave numberp = PressureRe = Reynolds numberTF = Transfer functionu, v, w = x-, y- and z-components of velocity, respectivelyU∞ = Free-stream velocityV , ~V = Velocity vector

∗Graduate Student, High Performance Computing Laboratory, Department of Aerospace Engineering.†Professor, High Performance Computing Laboratory, Department of Aerospace Engineering.

1 of 20

American Institute of Aeronautics and Astronautics

20th AIAA Computational Fluid Dynamics Conference27 - 30 June 2011, Honolulu, Hawaii

AIAA 2011-3238

Copyright © 2011 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

I. Introduction

Primitive variable (p, V )-formulations have been extensively used for solving both 2D and 3D incom-pressible Navier-Stokes equations. These formulations however suffer from the lack of an equation for theevolution of pressure. The solution of the momentum equations does not ensure updated solenoidal velocityfield at (n+ 1)th time-level even if the velocity field is solenoidal at nth time-level. This divergence-free con-dition of velocity is satisfied by various techniques in different variants of (p, V )-formulations. For example,in MAC1 or SMAC2 algorithm, this is done by solving a modified pressure Poisson equation that retains theerror in the divergence of the tentatively computed velocity field at (n+1)th time-level after time advancingthe momentum equation. Next, this velocity is projected to a solenoidal velocity field based on the com-puted pressure. In operator-splitting or fractional time-step methods, momentum equation is integrated in apredictor-corrector framework. In the predictor stage, an intermediate velocity field is computed by omittingthe pressure-gradient term, whereas in the corrector stage, mass conservation is enforced via the solutionof the pressure Poisson equation with solenoidality error of the intermediate velocity field acting as thesource term. But, (p, V )-formulations suffer from some serious problems, such as: (i) the pressure boundarycondition is ambiguous and complicated,3–5 (ii) in operator splitting method specification of wall-boundarycondition at the intermediate stage is not straight-forward6. Specification of wall-boundary condition inthe interim, is usually done by ad hoc extrapolation procedure from values obtained in previous time steps.This can provide numerical stability, however, at the cost of large splitting error7, which forces one to takesmaller time-steps. It has been shown that the correct way to implement these boundary conditions is quitememory-intensive8.

Problems faced by (p, V )-formulation can be avoided completely in derived variable formulations (either(ψ, ω)- or (V, ω)-formulation). In these formulations, pressure term is completely eliminated in the vorticitytransport equation -an evolution equation for vorticity. The boundary condition for vorticity is obtained ina straightforward manner, as compared to the pressure boundary condition in (p, V )-formulations. Vorticityboundary condition follows from the kinematic definition of vorticity9. Derived variable formulations arenot only accurate,9,10 but also convergence to steady state is faster as compared to primitive variableformulations11. From a physical point of view, Navier-Stokes equation for unsteady incompressible flowscan be viewed as an equation for the evolution of vorticity, a primary physical quantity, irrespective ofwhether the flow is laminar or turbulent. Primarily, vorticity is generated at physical boundaries for wall-bounded flow or due to flow instabilities in free shear layers. Hence the vorticity transport equation, iscentral for the analysis and solution of Navier-Stokes equation. Attendant velocity field can be obtainedfrom the solution of the Poisson equation for stream function (for 2D case) or by solving Poisson equationsfor the velocity components. It has been shown by Speziale12 for derived variable formulations that “all non-inertial effects (arising from both the rotation and translation of the frame of reference relative to an inertialframing) only enter into the solution of the problem through the implementation of initial and boundaryconditions in vorticity keeping the governing evolution equations unaltered”. In contrast, equations inprimitive variable formulation changes for non-inertial frames of reference, thereby altering it’s mathematicalcharacter altogether which might require a different algorithm to solve them.

Existence of the stream-function in 2D in (ψ, ω)-formulation guarantees exact satisfaction of divergence-free condition for velocity, which is not automatically satisfied in velocity-vorticity formulations. Extension of(ψ, ω)-formulation in 3D gives rise to vector potential-vorticity formulation. Vector potential in 3D cases hasthree components, which degenerates to a single component normal to the plane of flow in 2D cases (knownas the stream function). Velocity being the curl of vector potential, prescription of boundary conditionon vector potential is rather difficult13,14. In contrast, prescription of boundary condition for velocity isunambiguous and natural. Hence, among derived variable formulations, velocity-vorticity formulations aremore popular and easy to implement than vector potential-vorticity formulations.

II. Velocity-vorticity formulation in 2D

In 2D derived variable formulations, (ψ, ω)- and (V, ω)-formulations are often used. In (ψ, ω)-formulation,unknowns are the stream function and vorticity, whereas in (V, ω)-formulation, unknowns are u, v (x- andy-components of velocity respectively) and ω. For 2D flows, (V, ω)-formulation essentially consists of solvingthe evolution equation for vorticity and Poisson equation for the two-components of velocity. These governingequations in non-dimensional form in Cartesian co-ordinate, are given as15–18

2 of 20

American Institute of Aeronautics and Astronautics

∂ω

∂t+∂uω

∂x+∂vω

∂y=

1

Re

(

∂2ω

∂x2+∂2ω

∂y2

)

(1)

∂2u

∂x2+∂2u

∂y2= −

∂ω

∂y(2)

∂2v

∂x2+∂2v

∂y2=∂ω

∂x(3)

where the velocity ~V = (u, v) is expected to satisfy the divergence-free condition given by

∂u

∂x+∂v

∂y= 0 (4)

For simulation of 2D flows, staggered variable arrangement prescribed in Napolitano & Pascazio18 andGuj & Stella19 is adopted here, as shown in Figure 1. Staggered grid arrangement is preferred because erroris smaller in staggered grid, as compared to non-staggered grid20. It is noted that to evaluate convectivederivatives in staggered grids, it is required to interpolate u- and v-components at the location of thevorticity, as depicted in Figure 1. This is carried out by using an optimized version of the compact mid-pointinterpolation scheme21. This interpolation scheme is given as

α1fj−1 + fj + α1fj+1 =a1

2(fj− 1

2

+ fj+ 1

2

) +b12

(fj− 3

2

+ fj+ 3

2

) (5)

Here, fj ’s are the interpolated values at the jth-location, obtained from the known fj±n/2’s at the (j±n/2)th-

locations. To have a 4th order accurate scheme, a1 = 18(9 + 10α1) and b1 = 1

8(6α1 − 1), with α1 as a free

parameter. Additionally, α1 = 310

yields a 6th order accurate scheme for interpolation. However, we haveobtained α1 ≃ 0.41 by optimizing the integrated phase error of the scheme in the spectral plane to achievebetter dispersion relation preserving (DRP) properties. Using Fourier-Laplace transform one can express,

fj =∫

U(k) eikxj dk and fj =∫

U(k) eikxj dk. Resolution and performance of an interpolation scheme

can be characterized by a transfer function (TF) given by TF (k∆x) = U(k)/U(k), which is the ratio ofthe Fourier amplitude of the interpolated function to that of the original function. Ideally, one should haveTF (k∆x) = 1 for the whole range of k∆x upto the Nyquist limit. From Eq. (5), this transfer function isobtained as

TF (k∆x) =

[

a1 cos(k∆x/2) + b1 cos(3k∆x/2)

1 + 2α1 cos(k∆x)

]

We define an objective function given as

I =

∫ π

0

∣

∣

∣

∣

1 − TF (k∆x)

∣

∣

∣

∣

d(k∆x)

which is nothing but the phase error integrated over the whole range of k∆x. The TF given above is shownplotted as a function of k∆x in Figure 2(i) for indicated values of α1. The objective function I, is shownplotted in Figure 2(ii), as a function of α1. It is noted clearly that this objective function is minimum atα1 ≃ 0.41. This optimized fourth order scheme has been used here, for all reported computations.

III. Solution of 2D lid-driven cavity problem at super-critical Reynolds

number

Results are presented here for a 2D lid-driven cavity (LDC) problem for Re = 9000 and Re = 10, 000.Both these Reynolds numbers belong to post-critical range for 2D flow22,23. This problem has been solvedby employing two schemes, scheme A and scheme B, as described next.

Scheme A: In this approach, Eqs. (1) to (3) are solved, by first solving Eq. (1) for time advancing thevorticity field, followed by Eqs. (2) and (3) for u- and v-velocity components using BiCGSTAB24 iterativescheme. Thus, the solution obtained by this scheme will not identically satisfy solenoidality of the velocityfield as we stop the iteration procedure after the residue falls below a prescribed tolerance value of ǫ.

3 of 20

American Institute of Aeronautics and Astronautics

Scheme B: Here, the vorticity field is time advanced first, by solving Eq. (1). The u-velocity componentare obtained by solving Eq. (2), as before, using BiCGSTAB24 iterative scheme. The v-velocity componentis obtained directly by solving an equation derived by differentiating Eq. (4) with respect to y, which ensuressolenoidality of the velocity field. This equation is given as

∂2v

∂y2= −

∂2u

∂x∂y(6)

In both the schemes vorticity ω is updated first by 4th-order, four-stage Runge-Kutta scheme, whileconvective derivatives (i.e., ∂(uω)/∂x and ∂(vω)/∂y terms) are discretized using high accuracy optimizedcompact scheme, OUCS325 and dissipative terms (i.e., ∂2ω/∂x2 and ∂2ω/∂y2 terms) are discretized usingsecond-order central difference schemes. Here, either scheme A or scheme B is employed to obtain u- andv-velocity components.

A predetermined tolerance value of ǫ is used, while solving the above Poisson equations by BiCGSTABmethod. Maximum solenoidality error occurs inside the domain for scheme A and is directly dependentupon ǫ. In computing both the cases, a grid with (257 × 257) uniformly distributed points have been used.After every 100 time-steps, left-hand side of Eq. (4) (the solenoidality error) is computed and the maximum

absolute value is stored, using 2nd-order central difference scheme. In Figure 3(i), this ∇.~V error historyis plotted for Re = 9000 for both the schemes. Here, ǫ = 10−6 has been used. In Figures 3(ii) and 3(iii),vorticity value at the point, (x = 0.95, y = 0.95) is plotted as a function of time. Even though the level ofdivergence error shown in Figure 3(i), is orders of magnitude different from each other, yet it does not affectthe time of onset of physical instability, as shown in Figures 3(ii) and 3(iii). In Figure 4, vorticity contoursat indicated times are plotted for schemes A and B. Both the schemes show presence of a triangular vortex,as reported earlier for 2D supercritical LDC flow23,26,27. One notices a small time lag in motion of satellitevortices, at t = 1000, shown in the third frame of Figure 4(i) and 4(ii).

Next, we compute for a higher Reynolds number, i.e.. Re = 10, 000, results of which are shown in Figures5 and 6. Here, while employing scheme A, three different levels of ǫ = 10−4, 10−6 and 10−9 have been used.While solving Eq. (2) with scheme B, a tolerance value of ǫ = 10−6 has been used. From Figure 5(a), we cansee that maximum solenoidality error is markedly different for all the four cases, even when vorticity timehistory shown, at (x = 0.95, y = 0.95) in Figure 5(b), and vorticity contours in Figure 6, are almost similar.One also observes a small time lag for the satellite vortices at t = 1000 between scheme A with ǫ = 10−4,with other frames at this time in Figure 6. This time lag at t = 1000 among the vorticity contours obtainedby both scheme A with ǫ = 10−6, as well as, ǫ = 10−9 and that obtained by scheme B is not that prominent.

IV. Solving the excitation of zero pressure gradient flow past a flat plate

Next, zero pressure gradient (ZPG) flow past a semi-infinite flat plate is excited by simultaneous blowing-suction (SBS) exciter strip28,29 and flow field computed. A schematic diagram of the problem along withSBS exciter and appropriate boundary segments are shown in Figure 7(i). Origin of the co-ordinate systemis placed at the leading edge of the plate, whereas the computational domain starts slightly ahead of it, asshown in Figure 7(i). The SBS exciter prescribes time-dependent normal velocity vwall in between x = x1

and x = x2 at the wall as

vwall(x) = αFK Am(x) sin(βdt)

where, βd, Am(x) and αFK are the circular frequency, amplitude function of the excitation and amplitudecontrol parameter, respectively. The amplitude function, Am(x) is defined in a similar manner given in28 as

Am(x) =

a(

x−x1

xm−x1

)5

− b(

x−x1

xm−x1

)4

+ c(

x−x1

xm−x1

)3

for x1 ≤ x ≤ xm

−a(

x2−xx2−xm

)5

+ b(

x2−xx2−xm

)4

− c(

x2−xx2−xm

)3

for xm ≤ x ≤ x2

with a = 1.51875, b = 3.54375, c = 2.025 and xm = (x1 + x2)/2. For the presented results, we have usedαFK = 0.1. Simulations are performed by using scheme A. Additionally, another scheme, named as scheme

4 of 20

American Institute of Aeronautics and Astronautics

C, has been developed. In scheme C, after solving Eq. (2) for u-velocity, v-velocity is obtained by integratingEq. (4) as,

v(x, y) = v(x, 0) −

∫ y

0

(

∂u

∂x

)

dy (7)

Hence, scheme C ensures satisfaction of the divergence-free condition. For both the schemes A and C, avalue of ǫ = 10−6 is used. The used boundary conditions while solving this problem by Scheme A are asenlisted below:(i) At the inflow: u = U∞, ω = 0 and ∂v/∂x = 0 is prescribed. The last condition is obtained from thecontinuity equation at inflow.(ii) At the wall: u = 0, v = vwall is imposed. Wall vorticity is computed based on its kinematic definition.(iii) At the front part of the bottom boundary, which lies ahead of the leading edge of the plate: ∂u/∂y = 0,ω = 0 and v = 0 are prescribed.(iv) At the far-field boundary, as shown in Fig. 7(i): ω = 0, u = U∞ and ∂v/∂y = 0 are prescribed.(v) At the outflow, vorticity is calculated using convective Sommerfeld boundary condition given as,

∂ω

∂t+ Uc

∂ω

∂x= 0

with the artificial convective speed Uc chosen to be U∞. The condition used on u-component is ∂2u/∂x2 = 0,whereas a condition given as, ∂v/∂x = ω + ∂u/∂y is used for v-component of velocity.

While solving this problem of receptivity of the ZPG flow by Scheme C, above-mentioned boundaryconditions on vorticity and streamwise component of velocity are used. But the solution of the v-componentrequires only the prescribed wall-normal velocity at the wall. No condition on v- velocity component isrequired to be prescribed at the inflow, the far-field and the outflow.

Simulations have been performed with 1000 points in the streamwise and 400 points in the wall-normaldirections. Simulations retain the leading edge of the plate and a tangent-hyperbolic stretching is used tocluster points near the leading edge and wall. Vorticity contours at t = 20 after the onset of excitationobtained by using schemes A and C are plotted alongside that obtained by (ψ, ω)-formulation in Figures7(iii) to 7(v). In Figures 8(a), 8(b) and 8(c), disturbance u-velocity (ud) at three indicated heights are shown.This three heights correspond to (i) a point very close to the wall in Figure 8(a), (ii) a point at the edgeof the shear-layer in Figure 8(b) and (iii) a point far outside the shear-layer in Figure 8(c). The strikingsimilarity among these schemes is noteworthy.

V. Solution of flow inside 3D cubic cavity at supercritical Reynolds number

For 3D flow, the vorticity transport equation (VTE) in nondimensional form is given as,

∂~ω

∂t+ (~V · ∇)~ω = (~ω · ∇)~V +

1

Re∇2~ω (8)

The first term on the right hand side of Eq. (8) is the vortex stretching term, which was absent for 2D flows.

Using the vector identity ∇ × ( ~A × ~B) = ~A(∇ · ~B) − ~B(∇ · ~A) + ( ~B · ∇) ~A − ( ~A · ∇) ~B and divergence-free

condition on velocity and vorticity (∇ · ~V = 0 and ∇ · ~ω = 0), one gets the Laplacian form of VTE as

∂~ω

∂t+ ∇× (~ω × ~V ) =

1

Re∇2~ω (9)

The viscous term in the above Laplacian form can be further modified by noting the vector identity

∇2~ω = ∇(∇.~ω) −∇× (∇× ~ω)

which together with solenoidality condition, ∇ · ~ω = 0, yields the rotational form of VTE as

∂~ω

∂t+ ∇× ~H = 0 (10)

where, ~H = (~ω × ~V + 1Re∇ × ~ω) While, the computed vorticity field from Laplacian form is not strictly

non-solenoidal, but that computed from rotational form is. After updating the vorticity field using 4th-order

5 of 20

American Institute of Aeronautics and Astronautics

Runge-Kutta method using either Laplacian or rotational form of VTE, the attendant velocity vector iscomputed. The u- and v- components of the velocity vectors are calculated from the following Poissonequation given in vector form as,

∇2~V = −∇× ~ω (11)

The w-component is calculated directly from the continuity equation as,

∂2w

∂z2= −

(

∂2u

∂x∂z+

∂2v

∂y∂z

)

(12)

For this simulation also, we employ the staggered grid used in Wu et al.16 shown in Figure 9, on anelementary cell. The velocity components are defined at the center of the plane on which the component isperpendicular; while the vorticity components are placed at mid-location on the edges of the cube which areparallel to the direction of the vorticity components. Such grid staggering for 3D flows require additionalevaluation of derivatives of vorticity components for the vortex stretching terms. For example, the x-component of VTE would require evaluation of first derivatives of ωy and ωz at the location of ωx. Suchderivative evaluations are performed using a scheme which is a variation of the scheme given in Nagarajan& Lele21 for derivative evaluation in staggered grid, whose general interior stencil is given by

α2f′

j−1 + f ′j + α2f′

j+1 =b2

3∆x(fj+3/2 − fj−3/2) +

a2

∆x(fj+1/2 − fj−1/2) (13)

where f′

j ’s defined at the jth-location are the first derivatives of the known fj±n/2 function values at the

(j ± n/2)th-locations. For fourth order accuracy of the scheme one requires b2 = (22 α2 − 1)/8 and a2 =(9 − 6 α2)/8; with α2 = 9/62 yields a sixth order accurate scheme. However, for better DRP properties,one can optimize the resolution of the stencil given in Eq. (13) in the spectral plane with respect to theparameter α2.

Using Fourier-Laplace transform, one can define an equivalent wavenumber keq26,27 for the compact

scheme in Eq. (13) as

keq =2

∆x

[

a2 sin(k∆x/2) + (b2/3) sin(3k∆x/2)

1 + 2α2 cos(k∆x)

]

Ideally keq = k and therefore, one can define an objective function to be optimized with respect to α2 as,

Icomp =

∫ π

0

∣

∣

∣

∣

1 −keq

k

∣

∣

∣

∣

d(k∆x)

This objective function Icomp is shown plotted as a function of α2 in Figure 10(ii), while in Figure 10(i)we have plotted keq/k as a function of k∆x for different values of α2. It is noted from the figure that theoptimum is obtained for α2 = 0.216, which has been used here for the reported calculations.

We have computed the flow for the post-critical Reynolds number of Re = 3200 using a 80 × 80 × 80uniform grid. This Reynolds number is post-critical, as the computed results does not reach a steady stateeven after t = 2000. Experimental results of Koseff & Street30 also confirms this fact where flow inside alid-driven rectangular parallelopiped (with square cross section) of aspect ratio of L/D = 3 : 1 was studied.

Maximum values of |∇ · ~ω| within the computational domain, are plotted for the rotational and theLaplacian forms of velocity-vorticity formulation for Re = 3200 in Figure 11(i) and (ii), respectively. Wefind from Figure 11 that, maximum values of |∇ · ~ω| for rotational form is billion times smaller than that forLaplacian form, though levels of solution error remain bounded with time for these two forms.

In Fig. 12, computed velocity components in the mid xy-plane are compared with the experimentaltime-averaged velocity profiles reported in Koseff & Street30 for the same xy-plane. For comparison purpose,computed results are time-averaged between t = 1900 and t = 2000. The match is quite good, despite thefact that the computed results are for a cubic cavity, while the experimental results are for a rectangularcavity with square cross-section and aspect ratio of three. Results are also compared between rotational andLaplacian forms at t = 250 and t = 2000. Though the u-velocity profiles do not show any distinguishabledifferences at t = 250 and t = 2000, but there are differences for the v-velocity profiles at t = 250, specificallynear the rear wall. Also one notices that in frame (vi), v-velocity profile corresponding to the rotationalform has slightly higher maxima and slightly lower minima.

6 of 20

American Institute of Aeronautics and Astronautics

VI. Conclusion

Here, the importance of the solenoidality error in velocity is investigated for two different canonicalphysically unstable 2D flows in the context of (V, ω)-formulation. First one is the lid-driven cavity flow atsupercritical Reynolds number i.e., Re = 9000 and Re = 10, 000 and the second one is the zero pressuregradient flow past a semi-infinite flat plate excited by a SBS strip exciter to generate TS waves. We havedevised several schemes with respect to (V, ω)-formulation that exactly satisfy the continuity equation tosimulate these flows. Results obtained by employing these schemes are compared with that obtained byclassical (V, ω)-formulation where two Poisson equations for the velocity components are to be solved. Theeffect of the tolerance level of the BICGSTAB iterative solver on the evolution of vorticity and overall velocitysolenoidality error for solving the lid driven cavity flow at Re = 10, 000 by classical (V, ω)-formulation is alsoinvestigated.

The results strongly suggest that the solenoidality error in velocity does not at al lead to numerical flowinstabilities for external as well as internal flows. For internal flows this can cause only slight phase lagand nothing else whereas, for external flows, this does not play any role at al in determining physical flowinstabilities. This should be noted that conclusion is strictly applicable to (V, ω)-formulation and not to

(p, ~V )-formulation.We have also investigated the effects of vorticity solenoidality error in the context of 3D flows using

(V, ω)-formulation by solving the cubic LDC flow problem at Re = 3200, a post-critical Reynolds numberfor this flow problem. We have employed two different formulations to solve vorticity transport equation,(1) rotational form and (2) Laplacian form. The rotational form exactly preserves the solenoidality of thevorticity field provided the initial vorticity field is solenoidal. But the Laplacian form can not preservethe same. The nonsolenoidality error in Laplacian form creeps in due to the imposition of the boundaryconditions. In Figure 11, the vorticity solenoidality error is compared for these two different formulationsand shown despite having 109 higher magnitude between these two levels, they do not become unboundedeven after t = 2000. In Figure 12, the time averaged u- and v-velocity profiles at the mid (x, y)-plane iscompared with the experimental profiles of Koseff & Street30 for LDC flow with square cross section andaspect ratio three. Both the formulations give excellent match with the experimental results despite having109 magnitude difference in vorticity divergence error. Hence, it can be concluded from this simulation thatfor 3D flows time-averaged velocity profiles do not depend strongly on the solenoidality error of vorticity,even though instantaneous v-velocity profile exhibit mismatch particularly at earlier times of evolution. Acareful study of the flow field at Re = 3200 also reveals that rotational form captures details of the flow moreminutely. There is a necessity to simulate cubic LDC flow problem for higher Reynolds number to draw amore definitive conclusion.

In the present study, the ramifications of nonsolenoidality errors in (V, ω)-formulation for both 2D and 3Dflows are examined for physically unstable flows and shown that these do not lead to numerical instabilitiesper se, vis-a-vis (p, ~V )-formulation, where it is established that satisfaction of ∇· ~V = 0 is absolutely essentialeven for obtaining time-averaged flow-fields. This non-sensitive dependence on nonsolenoidal error for (V, ω)-

formulation gives it an edge over (p, ~V )-formulation and establishes itself as the most suitable candidate forperforming DNS of transitional/turbulent flows.

7 of 20

American Institute of Aeronautics and Astronautics

References

1Harlow, H. H. and Welch, J. E., “Numerical calculation of time-dependent viscous incompressible flow of fluid with free

surface,” Phys. Fluids, Vol. 8, 1965, pp. 2182–2189.2Kawamura, T., Takami, H., and Kuwahara, K., “A new higher-order upwind scheme for incompressible Navier-Stokes

equations,” Fluid Dynamic Res., Vol. 1, 1985, pp. 145–162.3Orszag, S. A., Israeli, M., and Deville, M. C., “Boundary conditions for incompressible flows,” J. Sci. Comp., Vol. 1,

1986, pp. 75–11.4Grescho, P. M. and Sani, R. L., “On pressure boundary conditions for the incompressible Navier-Stokes equation,” Int.

J. Numer. Meths. Fluids., Vol. 7, 1987, pp. 1111–1145.5Grescho, P. M., “On the theory of semi-implicit projection method for viscous incompressible flow and its implementation

via a finite element method that also introduces a nearly consistent mass matrix, Part I: Theory,” Int. J. Num. Meth. Fluids.,

Vol. 11, 1990, pp. 587–620.6Kleiser, L. and Schumann, U., Proceedings, Third GAMM Conference on Numerical Methods in Fluid Mechanics, Ed:

E. H. Hirschel , Vieweg, Braunschweig, 1980.7Perot, J. B., “An analysis of the fractional step method,” J. Comp. Phys., Vol. 103, 1993, pp. 51.8Mittal, R. and Balachander, S., “Direct numerical simulation of flow past elliptic cylinder,” J. Comp. Phys., Vol. 124,

1996, pp. 351.9Gatski, T. B., “Review of incompressible fluid flow computations using the vorticity-velocity formulation,” Applied

Numer. Math., Vol. 7, 1991, pp. 227–239.10Meitz, H. L. and Fasel, H., “A compact-difference scheme for the Navier-Stokes equations in vorticity-velocity formula-

tion,” J. Comp. Phys., Vol. 157, 2000, pp. 371–403.11Takemitsu, N., “Finite difference method to solve incompressible fluid flow,” J. Comp. Phys., Vol. 61, 1985, pp. 499–518.12Speziale, C. G., “On the advantages of the vorticity velocity formulation of the equations of uid-dynamics,” J. Comp.

Phys., Vol. 73, 1987, pp. 476–480.13E., W. and Liu, J.-G., “Finite difference methods for 3D viscous incompressible flows in the vorticity-vector potential

formulation on nonstaggered grids,” J. Comp. Phys., Vol. 138, 1997, pp. 57–82.14Wong, A. K. and Reizes, J. A., “An effective vorticity-vector potential formulation for the numerical simulation of

three-dimensional duct flow problems,” J. Comp. Phys., Vol. 55, 1984, pp. 98–114.15Gatski, T. B., Grosch, C. E., and Rose, M. E., “A numerical study of the 2-dimensional Navier-Stokes equations in

vorticity-velocity variables,” J. Comp. Phys., Vol. 48, 1982, pp. 1–22.16Wu, X. H., Wu, J. Z., and Wu, J. M., “Effective vorticity-velocity formulations for 3D incompressible viscous flows,” J.

Comp. Phys., Vol. 122, 1995, pp. 68–82.17Daube, O., “Resolution of the 2D Navier-Stokes equations in velocity-vorticity form by means of an inuence matrix

technique,” J. Comp. Phys., Vol. 103, 1992, pp. 402–414.18Napolitano, M. and Pascazio, G., “A numerical method for the vorticity-velocity Navier-Stokes equations in two and

three dimensions,” Comput. Fluids., Vol. 19, 1991, pp. 489–495.19Guj, G. and Stella, F., “A vorticity-velocity method for the numerical of 3D incompressible flows,” J. Comp. Phys.,

Vol. 106, 1993, pp. 286–298.20Huang, H. and Li, M., “Finite-difference approximation for the velocity-vorticity formulation on staggered and non-

staggered grids,” Comput. Fluids, Vol. 26, No. 1, 1997, pp. 59–82.21Nagarajan, S., Lele, S. K., and Ferziger, J. H., “A robust high-order compact method for large eddy simulation,” J.

Comput. Phys., Vol. 191, 2003, pp. 392–419.22Bruneau, C. H. and Saad, M., “The 2D lid-drivent cavity problem revisited,” Comput. Fluids, Vol. 35, 2006, pp. 326–378.23Sengupta, T. K., Vijay, V. V. S. N., and Singh, N., “Universal instability modes in internal and external flows,” Comput.

Fluids, 2010.24Vorst, H. A. V. D., “BiCGSTAB: A fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric

linear system,” SIAM J. Sci. Stat. Comp., Vol. 13, No. 2, 1992, pp. 631–644.25Sengupta, T. K., Ganeriwal, G., and De, S., “Analysis of central and upwind compact schemes,” J. Comp. Phys., Vol. 192,

No. 2, 2003, pp. 677–694.26Sengupta, T. K., Lakshmanan, V., and Vijay, V. V. S. N., “A new combined stable and dispersion relation preserving

compact scheme for non-periodic problems,” J. Comp. Phys., Vol. 228, 2009, pp. 3048–3071.27Sengupta, T. K., Vijay, V. V. S. N., and Bhaumik, S., “Further improvement and analysis of CCD scheme: Dissipation

discretization and de-aliasing properties,” J. Comp. Phys., Vol. 228, 2009, pp. 6150–6168.28Fasel, H. and Konzelmann, U., “Non-parallel stability of a flat-plate boundary layer using the complete Navier-Stokes

equation,” J. Fluid Mech., Vol. 221, 1990, pp. 311–347.29Sengupta, T. K., Bhaumik, S., Singh, V., and Shukl, S., “Nonlinear and nonparallel receptivity of zero-pressure gradient

boundary layer,” Int. J. Emerging Multidisciplinary Fluid Sci., Vol. 1, 2009, pp. 19–35.30Koseff, J. R. and Street, R. L., “On end wall effects in a lid-driven cavity flow,” J. Fluids Engg., Vol. 106, 1984,

pp. 385–398.

8 of 20

American Institute of Aeronautics and Astronautics

vω

ui-2,j-1/2

u

ui+2,j-1/2ui-1,j-1/2

vi-1/2,j-3

ui+1,j-1/2

ui,j+1/2

vi-3/2,j

vi-1/2,j+1

vi-1/2,j-1

vi-1/2,j-2

vi+3/2,jvi-1/2,j vi+1/2,j

ui,j-5/2

ui,j-1/2

ui,j-3/2

ωi-1,j

ωi,j-1

ωi+1,jωi-2,j ωi+2,j

ωi,j+1

ωi,j-3

ωi,j-2

ωi,j

Figure 1. Staggered representation of variables for two-dimensional (~V , ~ω)-formulation.

9 of 20

American Institute of Aeronautics and Astronautics

kh

TF

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

1.2

α 1 = 0.10α 1 = 0.15α 1 = 0.20α 1 = 0.25α 1 = 0.30α 1 = 0.35α 1 = 0.40α 1 = 0.45

α1increasing

(i)

α 1

I0 0.1 0.2 0.3 0.4 0.5

0

0.2

0.4

0.6

0.8

1 (ii)

α1 = 0.3

[α1]opt

Figure 2. (i) Transfer function for the staggered compact scheme for interpolation shown plotted as afunction of kh for different values of α1. (ii) The integrated phase error shown plotted as a function of α1.

10 of 20

American Institute of Aeronautics and Astronautics

time

max

(|∇.V

|)

0 200 400 600 800 100010-14

10-12

10-10

10-8

10-6

10-4

10-2

Re = 9000 ; Grid 256× 256(i)

Scheme BScheme A

time

ω

0 200 400 600 800 1000-9

-8

-7

-6

-5

-4

(ii) Scheme A

time

ω

0 200 400 600 800 1000-9

-8

-7

-6

-5

-4(iii) Scheme B

Figure 3. (a) Time history of maximum absolute divergence error obtained using schemes A and B plottedfor lid-driven cavity (LDC) flow for Re = 9000. (b, c) Time history of vorticity at (x = 0.95, y = 0.95)shown for schemes A and B, respectively.

11 of 20

American Institute of Aeronautics and Astronautics

(i) Scheme A

(ii) Scheme B

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1t = 500

x

y

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1t = 700

x

y

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1t = 1000

x

y

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1t = 500

x

y

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1t = 700

x

y

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1t = 1000

x

y

Figure 4. Vorticity contours for flow in LDC for Re = 9000 shown at indicated times for (i) scheme A and(ii) scheme B.

12 of 20

American Institute of Aeronautics and Astronautics

(a)

(b)

Time

max

|∇.V

|

0 500 100010-14

10-12

10-10

10-8

10-6

10-4

10-2

(ii) Scheme A withε = 10-6

Scheme A withε = 10-9

Scheme B withε = 10-6

Time

max

|∇.V

|

0 500 100010-14

10-12

10-10

10-8

10-6

10-4

10-2(i) Scheme A withε = 10-4

Time0 200 400 600 800 1000

-8

-7

-6

-5

-4 (iv) Scheme B withε = 10-6

ω

Time0 200 400 600 800 1000

-8

-7

-6

-5

-4 (iii) Scheme A with ε = 10-9

ω

Time0 200 400 600 800 1000

-8

-7

-6

-5

-4 (i) Scheme A withε = 10-4

ω

Time0 200 400 600 800 1000

-8

-7

-6

-5

-4 (ii) Scheme A with ε = 10-6

ω

Figure 5. (a) Time history of maximum absolute divergence error obtained using scheme A with indicatedvalues of ǫ and scheme B, plotted for flow in LDC for Re = 10000. (b) Time history of vorticity at (x = 0.95,y = 0.95) shown for (i) scheme A with ǫ = 10−4, (ii) scheme A with ǫ = 10−6, (iii) scheme A with ǫ = 10−9

and (iv) scheme B with ǫ = 10−6.

13 of 20

American Institute of Aeronautics and Astronautics

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1t = 300

x

y

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1t = 600

x

y

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1t = 1000

y

x

(i) Schema A withε = 10-4

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1t = 300

x

y

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1t = 600

x

y

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1t = 1000

xy

(ii) Scheme A withε = 10-6

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1t = 300

x

y

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1t = 1000

x

y

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1t = 600

x

y

(iii) Scheme A with ε = 10-9

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1t = 300

x

y

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1t = 600

x

y

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1t = 1000

x

y

(iv) Scheme B

Figure 6. Vorticity contours for flow in LDC for Re = 10000 shown at indicated times for (i) scheme Awith ǫ = 10−4; (ii) scheme A with ǫ = 10−6; (iii) scheme A with ǫ = 10−9 and (iv) scheme B with ǫ = 10−6.

14 of 20

American Institute of Aeronautics and Astronautics

xin xout

x

yInflow OutflowFar-field boundary

U∞

Exciter

x1x2

y = ymax

Edge of shear layer

Flat plate

(i)

Time

max

(|∇

.V|)

0 5 10 15 20 25

0.0006

0.0008

0.001

0.0012

0.00140.00160.00180.002 (ii) ( V - ω ) formulation: Scheme A

x

y

0 2 40

0.005

0.01

0.015

0.02

0.025

0.03 (iii) ( ψ - ω ) formulation

x

y

0 2 40

0.005

0.01

0.015

0.02

0.025

0.03 (iv) ( v - ω ) formulation: Scheme A

x

y

0 2 40

0.005

0.01

0.015

0.02

0.025

0.03 (v) ( v - ω ) formulation: Scheme C

Figure 7. (i) Schematic diagram for ZPG flow past a semi-infinite flat plate subject to simultaneousblowing and suction excitation shown with indicated boundaries. (ii) Maximum absolute divergence errortime-history shown for scheme A with ǫ = 10−6. Vorticity contour at t = 20 with exciter location markedby an arrow shown for (iii) (ψ, ω)-formulation; (iv) (~V , ~ω)-formulation with scheme A and (v)

(~V , ~ω)-formulation with scheme C.

15 of 20

American Institute of Aeronautics and Astronautics

0 2 4

-0.2

0

0.2

0.4(i) ( ψ − ω ) formulation

u d

x 0 2 4

-0.2

0

0.2

0.4

(iii) ( v - ω ) formulation

Scheme C

u d

x0 2 4

-0.2

0

0.2

0.4

(ii) ( v - ω ) formulation

Scheme A

u d

x

(a) y / δ*ex = 0.5

0 2 4-0.1

-0.05

0

0.05

0.1

0.15 (i) ( ψ − ω ) formulation

u d

x 0 2 4-0.1

-0.05

0

0.05

0.1

0.15 (ii) ( V - ω ) formulation

Scheme A

u d

x 0 2 4-0.1

-0.05

0

0.05

0.1

0.15 (iii) ( V - ω ) formulation

Scheme Cu d

x

(b) y / δ*ex = 6.75

0 2 4

-0.04

-0.02

0

0.02

0.04(i) ( ψ − ω ) formulation

u d

x 0 2 4

-0.04

-0.02

0

0.02

0.04

(ii) ( V - ω ) formulation

Scheme A

u d

x 0 2 4

-0.04

-0.02

0

0.02

0.04

(iii) ( V - ω ) formulation

Scheme C

u d

x

(c) y / δ*ex = 13.5

Figure 8. Disturbance u-velocity plotted as a function of streamwise distance at (a) y/δ∗ex = 0.5, (b)y/δ∗ex = 6.75 and (c) y/δ∗ex = 13.5. Here, δ∗ex is the displacement thickness at the location of the exciter.

16 of 20

American Institute of Aeronautics and Astronautics

z

y

x

w

u

v

ωy

ωx

ωy

ωy

ωz

ωx

ωx

ωzωz

(i, j, k)

(i, j, k+ 1)

(i+ 1, j, k)

(i, j+ 1, k)

Figure 9. Staggered arrangement of u-, v- and w-velocity components and ωx-, ωy- and ωz-vorticitycomponents shown on an elementary cell.

17 of 20

American Institute of Aeronautics and Astronautics

α 2

I com

p

0 0.1 0.2 0.3 0.4 0.50

0.2

0.4

0.6

0.8

1

α2 = 9 / 62

α2 = 0.216

(ii)

kh

keq

/k

0 0.5 1 1.5 2 2.5 30

0.4

0.8

1.2

1.6

2

2.4 α 2 = 0.10α 2 = 0.15α 2 = 0.20α 2 = 0.25α 2 = 0.30α 2 = 0.35α 2 = 0.40α 2 = 0.45

(i)

Figure 10. (i) Spectral resolution of the staggered compact scheme for evaluating 1st derivatives shown forindicated values of α2. (ii) Phase error plotted as a function of α2 for the stencil. Note that the optimumof Icomp is at α2 = 0.216 for the 4th-order version, as compared to α2 = 9/62 for the 6th-order accurateformula.

18 of 20

American Institute of Aeronautics and Astronautics

Time

max

(|∇.ω

|)

0 500 1000 1500 200096

96.5

97

97.5

98(i) Laplacian Form

Time

max

(|∇.ω

|)0 500 1000 1500 2000

10-10

10-9

10-8 (ii) Rotational Form

Figure 11. Flow inside a cubic LDC for Re = 3200. (i, ii) Time-history of vorticity divergence error plottedfor computations using Laplacian and conservative rotational forms of VTE, respectively.

19 of 20

American Institute of Aeronautics and Astronautics

u

y

-0.5 0 0.5 10

0.2

0.4

0.6

0.8

1

Experimental results ofKoseff & Street withL/D=3.0

Simulations with L/D=1.0

(i)

Simulation results are timeaveraged between t = 1900and t=2000

xv

0 0.2 0.4 0.6 0.8 1-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

Simulation with L/D=1.0Experiment with L/D=3.0

(ii)

Simulation results are timeaveraged between t = 1900and t=2000

u

y

-0.4 -0.2 0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

Laplacian FormRotational Form

(iii)

t = 250

x

v

0 0.2 0.4 0.6 0.8 1-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

Laplacian FormRotational Form

(iv)

t = 250

u

y

-0.5 0 0.5 10

0.2

0.4

0.6

0.8

1

Laplacian FormRotational Form

(v)

t = 2000

x

v

0 0.2 0.4 0.6 0.8 1-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

Laplacian FormRotational Form

(vi)

t = 2000

Figure 12. Flow inside a cubic LDC for Re = 3200. (i, ii) u- and v-velocity components at the mid-planeand time averaged between t = 1900 and t = 2000 obtained by using Laplacian form are compared with theexperimental time averaged results of Koseff & Street (1984). (iii, iv) u- and (v, vi) v-velocity componentsat mid-plane obtained by using rotational and Laplacian forms are compared at t = 250 and t = 2000, asindicated in the frames.

20 of 20

American Institute of Aeronautics and Astronautics