les and urans unsteady boundary layer strategies for pulsating and oscillating...

LES AND URANS UNSTEADY BOUNDARY LAYERSTRATEGIES FOR PULSATING AND OSCILLATING

TURBULENT CHANNEL FLOW APPLICATIONS.

Daniele Panara∗, Mauro Porta† and Thilo Schoenfeld†

∗DLR, Deutsches Zentrum fur Luft- und Raumfahrt e. V.Institut fur Verbrennungstechnik

Pfaffenwaldring 38-40, 70569 Stuttgarte-mail: [email protected]

web page: http://www.dlr.de/vt†CERFACS, Centre Europeen de Recherche et de Formation Avancee en Calcul Scientifique,

Avenue Gaspard Coriolis 42. 31057 Toulouse Cedex 01. France.e-mail: [email protected]

Key words: Fluid Dynamics, Turbulence, URANS, LES, Pulsating Flow, OscillatingFlow, Near Wall Modeling, Wall Shear Stress.

Abstract. The use of wall functions has been investigated for LES and URANS numeri-cal simulation in pulsating and oscillating channel flow applications. The results show thatthe wall function approach is accurate in the so called quasi-steady regime but there arediscrepancy with the experimental results in the intermediate frequency range. A specialattention is given to the wall-shear stress prediction, and in particular on the wall-shearstress phase shift with respect to the free stream velocity. In order to capture such un-steady flow effect, the boundary layer needs to be resolved. Different approach such as LowReynolds Number near wall turbulence modeling (URANS) or the proposed Wall-NormalResolved strategy (LES) seem to be suited for this purpose. The backdraw is unfortunatelythe increasing of computational points in the boundary layer and consequently the highercomputational cost.

1 INTRODUCTION

With the continuous growing of the computational resources, the use of unsteady nu-merical simulations is becoming a strategic tool in designing complex industrial compo-nents. Many examples of that can be found in turbomachinery and combustion chamberapplications. Commercial CFD softwares are widely used in industry and a critical factorfor the design development resides in an optimal balance between numerical costs andprediction accuracy. Unsteady Reynolds averaged Navier-Stokes (URANS) and, in thelast years, Large Eddy Simulations (LES) have been used increasingly by industry for thestudy of critical components. For internal turbulent flow simulations, in order to save

1

Daniele Panara, Mauro Porta and Thilo Schoenfeld

computational time, the wall-boundary layer resolution is often neglected. In both LESand URANS the tendency is to model the inner-part of the boundary layer by meansof so called wall functions. The use of wall functions significantly reduces the number ofcomputational points required in the boundary layer region and for this reason it is widelyused. However, it is questionable whether in pulsating and unsteady flow conditions thehypothesis under which the wall functions have been developed are still valid. A deeperinvestigation on the accuracy of the use of wall functions for unsteady turbulent flowapplications is therefore needed.

Despite their simple geometry, pulsating channel flows are representative of many in-teresting industrial configurations. In the present work, the accuracy of the use of wallfunctions in LES and URANS is investigated by means of such testing cases. Our at-tention is focused on the unsteady wall-shear stress prediction since it is important alsoas an indirect measure of the unsteady wall heat transfer. A correct evaluation of themean and peak values of the unsteady wall heat transfer is critical in combustion chamberapplications and in general for situations where the thermal resistance of the materials isof concern.

In the present work two very different CFD research solvers are used. OpenFoam1 hasbeen chosen for the URANS calculations and AVBP2 for the LES calculations. The twocode differ not only for the discretisation schemes used but also for the mathematicalformulation of the problem. A brief description of the codes and governing equations willbe given in the next sections. For the near wall treatment, Both codes employ a similarwall function approach.A brief description of the wall-function approach is reported insection 2. Details about the LES code and its wall functions implementation are given insection 3 and 3.1. In section 3.2 a validation of the LES code in an oscillating channelflow configuration against DNS results from Spalart and Baldwin3 is presented. Detailsof the URANS solver are given in section 4 and the implementation of wall function isbriefly described in section 4.1. The URANS code is validate against the experimentalresults of Tardu et al.4 in section 4.2. Finally in the last part of the paper, the URANSresults obtained in section 4.2 for pulsating flow conditions are compared with compa-rable LES pulsating channel simulations and experimental results using characteristicnon-dimensional parameters. Main attention is paid as already emphasized on wall-shearstress and its phase shift whit respect to the center-line velocity.

2 The Law of the Wall

The effect of solid boundaries on viscous turbulent fluid flows resides in the so calledboundary layer. Outside the boundary layer the so called free-stream velocity (Ue) can becomputed as the flow would have been considered in-viscid. The boundary layer thickness(δ) is defined as the distance from the wall where the fluid velocity is 0.99Ue. Inside theboundary layer we can distinguish two sub-layers, the so called inner and outer layer.Indicating with y the normal distance to the wall, the outer layer resides generally inbetween 0.2δ and δ. The outer layer is affected by the free-stream velocity and pressure

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gradients. The inner layer instead has a more universal character and can be decomposedin three sub-layers: the logarithmic region, the buffer layer and the linear sub-layer.Almost all turbulent flows at finite wall shear stress exhibit in the inner layer an universalvelocity distribution general referred as the universal law of the wall which is describedin the following.

If we non-dimensionalise the velocity and wall distance as follows:

y+ =uwy

νu+ = u

uw(1)

where uw is the skin friction velocity,

uw =

√τw

ρ(2)

we observe that the linear sub-layer generally resides between the wall and values ofy+ close to 5, the buffer layer between values of y+ around 5 and 40 and the logarithmicregion above values of y+ equal to 40. The velocity profiles in the linear sub-layer and inthe logarithmic layer can be analytically expressed as follows

u+ = y+ 0 < y+ < 5 (3)

u+ =1

κln y+ + C y+ > 40 (4)

Due to the general validity of 4 and in order to save computational time, the viscouseffect of the wall is often modeled imposing at the wall boundary a wall-shear stress (τw)extracted from the application of 4 to the first numerical control volume. This approachis often referred as the wall-function approach and it is the object of our .

3 LES Solver

DNS and LES computations, object of this work, were performed using the AVBP codedeveloped by CERFACS and IFP. AVBP is a parallel CFD code that solves the laminarand turbulent compressible Navier-Stokes equations in two and three space dimensions onunstructured and hybrid grids. The data structure of AVBP employs a cell-vertex finite-volume approximation. The basic numerical methods are based on a Lax-Wendroff or aFinite-Element type low-dissipation Taylor-Galerkin discretisation in combination with alinear-preserving artificial viscosity model. In this paper only the former was used, thestudy of the influence of the numerical scheme on the solution is left for future works. Thetime discretisation is explicit and makes use of a Runge-Kutta multi-stage time stepping.For turbulent compressible flows, AVBP solves the LES formulation of the Navier-Stokesequations. According to the LES approach, the governing equation are filtered in spacebefore discretising and solving. Additional unresolved terms appears in the convectivefluxes:

τijt = −ρ(uiuj − uiuj) (5)

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for the Reynolds stresses and

qit = −ρ(uiE − uiE) (6)

for the heat flux vector. The over-bar represent a filtered numerically resolved quantityand the tilde represents a mass-weighted Favre filtering such as:

ρf = ρf (7)

ρ,E and ui represent respectively the density, internal energy and the ith componentof the velocity vector u. The unresolved sub-grid scale (SGS) terms are generally closedusing the following formulation:

τijt = 2ρνtSij −

1

3τll

tδij (8)

for the Reynolds stresses and

qit = −λt

∂T

∂xi+

N∑

k=1

Ji,kths,k (9)

for the heat flux vector. Where

λt =νtρCp

P tr

(10)

and

Sij =1

2(∂ui

∂xj+

∂uj

∂xi) − 1

3

∂uk

∂xkδij (11)

Since P tr is assumed to be constant and equal to 0.9 the closure of the SGS terms is

shifted to the determination of νt. In our computations two different closure are used:the classical Smagorinsky model5

νt = (CS∆)2√

2SijSij (12)

and the WALE model6

νt = (Cw∆)2 (sdijs

dij)

3/2

(SijSij)5/2 + (sdijs

dij)

5/4(13)

where Cw and CS are model constants ( Cw = 0.4929 and CS = 0.18 ), ∆ is thecharacteristic filter length and

sdij =

1

2(g2

ij + g2ji) −

1

3g2

kkδij (14)

gij denotes the resolved velocity gradient. The WALE model was developed for wallbounded flows in an attempt to recover the scaling laws of the wall without using the wallfunction approach.

4


Figure 1: Typical velocity profile near the wall and notation used for near-wall quantities.

3.1 Wall Functions Implementation

The wall law implemented in AVBP is described in Schmitt9 and will be here shortlydescribed. As mentioned above AVBP uses a cell-vertex scheme. All quantities arethen stored at the cell-corners. For the calculation of the viscous fluxes AVBP needsalso the shear stresses at the cell boundary and heat fluxes. Imposing the appropriatevalue of velocity and temperature at the boundary plus the correct fluxes using the wall-law formulations constrains the flow too much and leads to oscillatory solutions. Thestrategy used by Schmitt is then to impose the wall-shear stress τw (and heat flux qw) atthe boundary using the wall-function approach without fixing the value of velocity andtemperature at the cell corners (u1 and T1 in figure 1). Only the normal component ofthe velocity at the wall is imposed to vanish at the wall for continuity reasons. This isequivalent as showed in figure 1 to imagine the real wall boundary shifted by a smalldistance δw away from the computational domain. Assuming that the shift is smallcompared to the distance between the wall and the first point in which the wall-functionis evaluated (δw ≪ yw), it can be neglected when computing the wall distance. The wallshear stress is then imposed at the boundary following the the steps in table 1.

3.2 LES code Validation for Oscillating Flow

In order to validate the code in a turbulent oscillating channel flow application thenumerical data are compared with a DNS (Direct Numerical Simulation) from Spalart andBaldwin3. Two series of near wall treatments have been employed: the Wall FunctionApproach as explained in section 3.1 and a so called Wall-Normal Resolved approachwhich details are given in the following.

The computational domain consists of a cubic box of 0.006m side centered in the axisorigin. On the upper and lower box-sides solid wall boundary conditions are applied. Theother remaining boundaries are treated as periodic. The flow is considered aligned with

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Step 1 Compute uτ iteratively from Eq. (3) or (4) with Eq. (1).Input values: u = u2, ν = ν(T1), y = ∆y.

Step 2 Compute τw from Eq. (2) with ρ = ρ1.

Step 3 Apply τw and advance flow equations.

Step 4 Set normal component of u1 to zero and go to Step 1.

Table 1: Working principle of the wall-function boundary condition.

Fine Grid Coarse GridWall-Normal

Resolved 31x31x31 21x21x21Wall-Functions 37x15x37 31x11x31

Table 2: grid points and reference directory table

the x axis. The y axis is normal to the walls and the z axis is aligned with the span-wisedirection.

The flow is oscillated using a pressure source term which realize an harmonic pressuregradient:

−1

ρ

∂p

∂x= K sin(ωt) (15)

For the Wall-Function computations two equally-spaced grid in the three axis directionare used with different refinement. A wall function treatment is employed in the near wallregion as explained in 3.1.

In the Wall-Normal Resolved computations, the used grids are equally spaced in thex and z directions, but stretched in the wall-normal direction in order to resolve theboundary layer up to a y+ value of the order of 2. The y+ value is computed using in firstapproximation the usual steady channel flow correlations and the maximum velocity valuein the cycle. An hyperbolic tangent stretching law has been employed and two differentgrid refinement are presented. The number of grid points used for each computation aresummarized in the table 2.

The fluid considered is N2 with a kinematic viscosity value of 1.710−5m2/s. The max-imum τw is calculated considering the maximum cycle velocity at the channel center line.We choose a maximum velocity amplitude, Umax ≈ 70m/s and a frequency of 100Hz inorder to have a Reynolds number based on the Stokes lenght around 1000 and comparablewith the results in Spalart and Baldwin3. The Reynolds number and the Stokes lenghtdefinition are reported below:

ω = 2πf = 628rad/s (16)

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δl =

√2ν

ω= 2.327 · 10−4m (17)

Reδl=

Umaxδl

ν≈ 1000 (18)

The pressure gradient source term was in first approximation evaluated using the an-alytical laminar results.

u(y, t) = −K

ωeiωt{1 −

cosh(y√

iων)

cosh(h2

√iων

)} (19)

In order to have an oscillating velocity at the center line as

Uc(y, t) = Umaxsin(ωt) (20)

we observed from 19 that K should be set as

K = ωUmax (21)

For large values of ω as reported in Schlichting8 19 become:

u(y, t) =K

ω[cos(ωt) − e

√ω2ν

(h2−y) cos(ωt−

√ω

2ν(h

2− y))] (22)

and the fluid in the core oscillates inviscid, with a phase shift of 90 degree respect tothe pressure source term. We initialized than the flow with an uniform velocity equal to−Umax and a white noise with typical value of u′ = 0.02Umax.

For the Wall-Resolved computations, we used the WALE SGS model as described in thesection 3 in order to reproduce the asymptotic behavior of the SGS turbulence term to thewall for bounded flows6. For the Wall-Functions computations, we used instead the usualhomogeneous smagorinsky SGS model together with the wall function implementationdescribed in section 3.1.

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3.2.1 Results

The numerical results for the Wall-Function and Wall-Normal Resolved computationsare reported in figure from (3.2.2) to (3.2.2) and from (3.2.3) to (3.2.3). The resultdiscrepancies between coarse and fine grid where negligible and for this reason only thefine grid results are reported. For each time phase (π/6, π/3, 2π/3, 5π/6 and π) thenon-dimensional values of the wall shear stress (τw/U2

o ), the velocity distribution (u/Uo),the fluctuation of the velocity components (u′/Uo,v

′/Uo and w′/Uo) and thedimensionalpressure fluctuations (P ′) are reported in function of the non-dimensional wall distance.The value of flow quantities have been non-dimensionalised using the velocity amplitude(Uo) and the stokes lenght δl.

The results are obtained by phase-locking averaging the numerical instantaneous valuesover several cycles. As we can see from the graphs, the velocity profile are quite wellcaptured by both near wall approaches. We noticed a non-zero value of the velocityat the wall due to the wall-function implementation as explained in section 3.1. In theWall-Function computations, the value of the velocity fluctuations are not well capturedat the wall as we expected for the reason that near the wall the turbulent structure arenot resolved since they are indeed modeled by the wall law. The Wall-Normal Resolvedcomputations instead, reproduce quite well the turbulent structure at the wall even if thegrid is resolve only in the y-direction. The values of u′/Uo,v

′/Uo and w′/Uo agree verywell with the DNS results. Concerning τw we observe a good agreement in both series ofcomputations. We notice that the Wall-Normal Resolved computations predict pressurefluctuations one order of magnitude larger respect to the Wall-Functions computations.In figure (3.2.3) τw is reported in phase and compared with the DNS results. As we cansee the Wall-Function computations seem to reproduce better the amplitude of the wallshear stress oscillations. What we notice is a little phase shift between the peak value ofτw that is not present in the Wall-Normal Resolved results. The Wall-Normal Resolvedτw predictions seem to underestimate the peak value of the wall-shear stress but seemto be more in phase with the DNS data. The overall behavior of both approach is quitegood in this oscillation frequency.

The most striking discrepancy is on the pressure fluctuation prediction. Unfortunatelyno direct DNS data are available. In figure (3.2.3) the pressure term of the Reynolds-stressbudged is reported compared with the DNS data. The figure shows that the Wall-NormalResolved result agree with the DNS data. The Wall-Function results are not reported inthe graph since out of scale. The results oscillations are probably due to numerical noisedue to the discretisation scheme employed.

Pterm = Π12 = −1

ρ

(u′∂P ′

∂y− v′∂P ′

∂x

)(23)

8


3.2.2 Wall-Function Computations

Figure 2: Phase π/6, Fine grid, Wall-Function Computation.

Figure 3: Phase π/3, Fine grid, Wall-Function Computation.

9


Figure 4: Phase 2π/3, Fine grid, Wall-Function Computation.

Figure 5: Phase 5π/6, Fine grid, Wall-Function Computation.

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Figure 6: Phase π, Fine grid, Wall-Function Computation.

3.2.3 Wall-Normal Resolved Computations

Figure 7: Phase π/6, Fine grid, Wall-Normal Resolved Computation.

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Figure 8: Phase π/3, Fine grid, Wall-Normal Resolved Computation.

Figure 9: Phase 2π/3, Fine grid, Wall-Normal Resolved Computation.

12


Figure 10: Phase 5π/6, Fine grid, Wall-Normal Resolved Computation.

Figure 11: Phase π, Fine grid, Wall-Normal Resolved Computation.

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Figure 12: Wall-shear stress vs. phase, Fine grid, Wall-Normal Resolved and Wall Function Computation.

Figure 13: Pressure term

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4 URANS Solver

The URANS calculation where performed using the OpenFoam solver. OpenFoam(Open Field Operation and Manipulation) is a CFD toolbox that uses finite volume nu-merics to solve systems of partial differential equations ascribed on any 3D unstructuredmesh of polyhedral cells. OpenFOAM is designed to be a flexible, programmable envi-ronment for simulation by having top-level code that is a direct representation of theequations being solved. The top-level code used for our computations is the standardOpenFoam solver turbFoam, a transient solver for incompressible, turbulent flow thatwill be shortly described below.

turbFoam solves the URANS equation for a turbulent fluid flow using a robust, implicit,pressure-velocity, iterative algorithm based on the PISO scheme7 (Pressure-Implicit withSplitting of Operators). In the following we will briefly describe the URANS governingequations together with the k-epsilon turbulence model formulation and we will give alsodetails on the PISO scheme.

4.0.4 Governing Equations

The momentum and continuity equations for an incompressible fluid can be written as:

∂ui

∂t+

∂ujui

∂xj

= −1

ρ

∂p

∂xi

+∂τij

∂xj

(24)

and

∂ui

∂xi= 0 (25)

For a Newtonian fluid the shear stress tensor τij can be expressed as:

τij = ν(∂ui

∂xj

+∂uj

∂xi

) − 2

3νδij

∂uk

∂xk

(26)

The above equations are valid for laminar flows and are also able to predict turbulentflows if a sufficient grid resolution is used (DNS computations). When the grid resolutionis not enough to meet the requirements of DNS calculations, a generic fluid quantity canbe decomposed as follows:

f(t, x) = f(x, t) + f ′(x, t) (27)

It means that an instantaneous value of the fluid characteristic f can be decomposedin the summation of its ensemble average value f and its turbulent fluctuations f ′ wherethe ensemble averaging operator can be written as:

f(t, x) = limn→∞

(1

n

n∑

k=1

fk(x, t)) (28)

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fk represents, physically, the value of the quantity f at time t and at position x duringthe kth experiment realization. Mathematically fk represents a random value of f . Theprobability density function of f depends on the flow turbulence. The Ensemble averagingoperator filters out the contribution of the turbulence fluctuations and the governingequations can be then rewritten just in terms of the averaged variables:

∂ui

∂t+

∂uj ui

∂xj= −1

ρ

∂p

∂xi+

∂τij

∂xj− ∂

∂xj(u′

iu′j) (29)

and

∂ui

∂xi= 0 (30)

A new unknown appears in the set of equations: the turbulence stress tensor

τt = −u′u′ (31)

The modeling of turbulence stress tensor generally make use of the Boussinesq hypoth-esis as follows:

τt = νt(∂ui

∂xj+

∂uj

∂xi) − 2

3νtδij

∂uk

∂xk(32)

The system closure is then shifted on the νt definition. In the URANS computationsa k-ǫ turbulence model has been used in its Low Reynolds (Launder and Sharma11) andHigh Reynolds Number formulation. In both cases the turbulent kinematic viscosity, νt

is expressed as

νt = Cµk2

ǫ(33)

where k and ǫ represent the turbulent kinetic energy and its dissipation rate and areobtained by solving two additional partial differential equations.

In the most general case we can write the transport equations for k and ǫ as

∂k

∂t+ ui

∂k

∂xi

= P − ǫ +∂

∂xi

[(ν +νt

σk

)∂k

∂xi

] (34)

∂ǫ

∂t+ ui

∂ǫ

∂xi= Cǫ1f1

ǫ

kP − Cǫ2f2

ǫ2

k+

∂

∂xi[(ν +

νt

σǫ)

∂ǫ

∂xi] + E (35)

ǫ = ǫ + D (36)

νt = Cµfµk2

ǫ(37)

Where Cµ, Cǫ1, Cǫ2, σk, σǫ, are constants and f1, f2, fµ, D and E have different formu-lations depending on the specific Low Reynolds Number turbulence model employed. TheLow Reynolds Number formulation differs from the High Reynolds Number formulation

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D Cµ Cǫ1 Cǫ2 σk σǫ fµ f1 f2 E

2ν(∂√

k∂y

)2 0.09 1.44 1.92 1.0 1.3 e[ −3.4

(1+Rt/50)2)]

1.0 1 − 0.3e−R2t 2ννt(

∂2u∂y2 )2

Table 3: Constants and functions for the k-ǫ Low Reynolds Number model of Launder and Sharma11

by the inclusion of the viscous diffusion terms and of the functions f . Also the D and Eterms are added in some cases to better represent the near-wall behavior. For a completereview on Low Reynolds Number turbulence model and their applications to unsteadyflow we refer to Fan9 and Patel10. In general the term f2 is introduced to incorporate lowReynolds effect found experimentally in the decay law of turbulence. The terms f1 D andE are introduced to recover the behavior of k and ǫ at the wall for steady flows. The termD in fact, is introduced to take into account the non-zero value of epsilon at the wall.The diffusion terms in general introduce more physics in the transport equations and theterm fµ is the most important term that should model the viscous and pressure straineffects. In general the term fµ takes into account the fact that the turbulent viscositycannot be modeled just as a scalar field but in complex flow conditions has more likely atensor field character which can take better into account the anisotropy of the flow due tounsteadiness and curvature-pressure gradient effects. In the computations performed theLow Reynolds k-ǫ model from Launder and Sharma11 has been used. Model Functionsand constants are reported in table

4.0.5 PISO Scheme

In the following a brief description of the PISO scheme is given. Since the density isassumed to be constant we can consider for simplicity instead of the pressure a correctedpressure quantity: p = p/ρ and we can proceed to the finite volume discretisation ofequation 29 and 30 as follows:

1

δt(un+1

i − uni ) = H(un+1

i ) − ∆ipn+1 (38)

and

∆iun+1i = 0 (39)

Here we refeer to the notation of Issa7 and for simplicity the over bar has been omitted.The operator H stands for the finite-volume representation of the spatial convective anddiffusive fluxes of momentum. ∆i represent instead the finite-volume discretisation of thegradient operator. The H operator is of course not linear but we assume that dependingon the discretisation used and on the flux formulation it can be linearized in the generalform

H(ui) = Amui,m (40)

where the suffix m is a grid node identifier that corresponds to the stencil used for thedefinition of the spatial fluxes at the finite-volume surface.

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A pressure equation can be obtained from 38 expliciting the p term and operating againthe divergence:

∆2i p

n+1 = ∆iH(un+1i ) +

∆iuni

δt(41)

Note that the term∆iu

n+1i

δtdoes not appear in the above equation because of 39. The

pressure equation indeed satisfies the continuity equation also. According to the PISOscheme the solution can be sought using a series of predictor corrector steps as follows:

1. Assuming pn and uni a predicted value of un+1

i ( u∗i ) can be obtained solving

1

δt(u∗

i − uni ) = H(u∗

i ) − ∆ipn+1 (42)

2. Assuming u∗i and un

i a predicted value of pn+1 ( p∗ ) is obtained by

∆2i p

∗ = ∆iH(u∗i ) +

∆iuni

δt(43)

3. Assuming the predicted values, u∗i and p∗ for un+1

i and pn+1 a corrected value for ui

(u∗∗i ) can be derived again from the momentum equation:

1

δt(u∗∗

i − uni ) = H(u∗

i ) − ∆ipn+1 (44)

Because of 43 u∗∗i satisfies already the zero divergence condition. If we apply the

divergence operator to 44 in fact we obtain

1

δt(∆iu

∗∗i − ∆iu

ni ) = ∆iH(u∗

i ) − ∆2i p

n+1 (45)

and using 42 with 45 we recover the discretised continuity equation for u∗∗i :

∆iu∗∗i = 0 (46)

4. steps 2 and 3 can be iterate in order to get more accurate values of pn+1 and un+1i .

However it is shown by Issa7 that p∗∗ and u∗∗∗i approximate the exact solution

sufficiently well for most practical purposes.

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In the following the numerical discretisation used in turbFoam for the simulations,which are object of this work, are described. As mentioned above turbFoam solves themomentum equation in 24 using the PISO scheme. We give here more details about thediscretisation of the convective and diffusive terms in H . The nonlinear convection term( ∂ujui

∂xj) is integrated over a control volume and linearized as follows:

∫

V

∂ujui

∂xj

dV =∫

V∇ • (~u~u)dV =

∑

f

Sf • ~uf~uf = ap~up +∑

N

aN~uN (47)

where the ap and aN are functions of ~u, the velocity vector. The suffix f indicates theinterpolate quantity on the control volume boundary. The suffix p refers at the velocityat the centroid of the control volume and the summation over N is intended over the Nneighbor points that surround the considered control volume V . The coefficient dependson the differencing scheme used. In our case we have used a blended differencing schemethat combines the accuracy of the central differencing and upwind differencing schemes.The scheme is second order accurate and bounded. The nonlinear diffusion term

∂τij

∂xj

= −∇2(νeff~u) −∇ • [νeff(∇~uT − 2

3Tr(∇~uT ))] (48)

is discretised explicitly assuming the turbulent viscosity from the resolution of theadditional turbulent equations and the velocity distribution at the time step n. Afterintegration over the control volume, the laplacian term is calculated using a Gauss linearcorrected scheme ( unbounded, second order and conservative) and the divergence termusing a simple Gauss linear scheme ( unbounded and second order accurate ). Detailsabout the definition of νeff depending on the turbulence model are given in the nextsection. After discretisation the PISO pressure equation is solved using an ICCG (In-complete Cholesky - Conjugate Gradient) solver and the momentum equation is solvedusing a BICCG (Bi-Convolution Conjugate Gradient) solver. Two PISO corrector stepsare used.

4.1 Wall Functions Implementation

The use of wall functions is based on two important assumptions:

1. The validity of the universal law of the wall (4)

2. The assumption of turbulent local equilibrium in the near wall region:

P = ǫ; P = νt(∂u∂y

)2 (49)

In the above hypothesis, assuming known the distribution of k and ǫ near the wallby solving their transport equation, it is possible to obtain an expression for τw thatcan be used as boundary condition for the solution of the momentum equation. Forthe solution of the k and ǫ transport equations is assumed

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∂k

∂y

∣∣∣ = 0 (50)

and

ǫ|w = 0 (51)

If indicate with the subscript p the flow variables at the first grid point Pand weassume that the point is situated in the log-law region, the expression for τw at theboundary is given by

τw =κCµρupk

1/2p

ln( 1µEρypk

1/2p C

1/4µ )

(52)

The above expression comes from the mentioned hypothesis if we follow the followingsteps.

From 49 we have

ǫ = νt∂u

∂y

∂u

∂y(53)

using the definition of τw and uτ :

τw = µt∂u

∂y; uτ =

√τw

ρ(54)

we can obtain the following for ǫ:

ǫ = u2τ

∂u

∂y(55)

Deriving respect to y the wall law we have

∂u

∂y=

uτ

κy(56)

and 55 can be expressed

ǫ =u3

τ

κy. (57)

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Another expression for ǫ can be sought from 53 using the definition of νt for theHigh Reynolds Number k-ǫ model (νt = Cµ

k2

ǫ) and 56:

ǫ = Cµk2

ǫ

u2τ

(κy)2(58)

and finally from 57 and 58 we obtain

uτ = C1/4µ k1/2 (59)

Using 59 in 4 it is easy to recover 52.

4.2 URANS code Validation for Pulsating Flow

The URANS code is validated against the experimental data of Tardu et al.4. The testchannel is 100 mm in width, 2600 mm in length and 1000 mm in span. The working fluidis water and the flow is pulsated using a special device which details are given in 4. Theflow can be considered isothermal.

Due to the pulsation, the phase locked average value of the velocity at the channel axiscan be expressed as

uc(t) = Uc(1 + auc cos ωt) (60)

or

uc(t) = Uc + Auc cos ωt (61)

The values of auc are varied in the experiment in the range from 0.1 to 0.6. The valueof Uc varies up to 0.5 m/s. The pulsation frequency is given in relation to the so calleddimensionless viscous Stokes layer thickness (l+s ) defined below

l+s =

√2

ω+(62)

with

ω+ =ων

u2w

(63)

Where u2w is the skin friction velocity of the relative steady channel case with Uc velocity

at the center line.Also for the URANS simulations, two series of near wall treatments have been em-

ployed: the Wall Function Approach as explained in section 4.1 together with a k-ǫHigh Reynolds Number model and a resolved boundary layer approach using the k-ǫ LowReynolds Number model from Launder and Sharma11.

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The grid deployed with Low Reynolds Number turbulence models ( LR-Grid ) is a twodimensional grid with 150x80 points. The points in the direction normal to the channelwall are stretched using a simple grading algorithm in which the ratio between the largerand the smaller cell has been set to 5. The instantaneous values of y+ are varying duringthe unsteady computations but are always between 10 and one.

The grid deployed with wall functions ( HR-Grid ) is not stretched and it is composedof 150x40 points. As expected the Wall Functions approach requires a minor number ofcomputational points and the y+ values are in this case of the order of 50.

Concerning the boundary conditions, a turbulent inlet profile is pulsated at the inletand a fixed static value of pressure is prescribed at the outlet.

uinl = umax(2y

h)

17 (64)

The value of the turbulent kinetic energy k is prescribed at the inlet using the experi-mental value on u′u′ reported by Tardu et al4. The value of the inlet dissipation rate of theturbulent kinetic energy has been estimated using the following mixing length hypothesis:

ǫ =C0.75

µ k1.5

δ(65)

where δ for fully developed pipe flows can be taken as half channel width. As noticedby Tardu et al in4, the channel length is only 52 times the channel width and rather shortto ensure fully developed turbulent flow. However due to the unsteady regime the flow hasbeen observed to be well established already at x/h = 42. According to our observation,no alteration in the resulting flow field occurs due to small changes of turbulent inletparameters.

4.2.1 Results

In figure 4.2.1 we report the near-wall velocity profiles in different phases comparedwith the experimental results for l+s = 8.1,auc = 0.64 and Uc = 0.169 m/s. The computedskin friction velocity for the steady case is 0.92 cm/s. According to the definition of l+sit is possible to calculate the pulsating flow frequency: f = 0.41 Hz. The high Reynoldsnumber and low Reynolds number boundary layer treatments are indicated with HR andLR. The figure shows the first limitation of the wall law approach. In this flow regime,the magnitude of the pulsations determines the flow reversal close to the solid walls. Thisis not captured by the HR model even though there is quite a good agreement betweenthe HR and LR model far from the wall.

If we compute now the wall shear stress using 52 for the HR case and the following forthe LR model:

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Figure 14: Instantaneous velocity profiles in the presence of reverse flow. Uc = 16.9 cm/s, auc = 0.64,l+s

= 8.1.

τw = ρν∂u

∂y|w (66)

we obtain a phase shift with respect to the velocity centreline of −8◦ and 33◦ as itis shown in figure 6 . Besides the calculations done for l+s = 8.1, a variety of furthersimulations were done for other values of l+s . In figure 6 the value of wall-shear stressphase shift is shown as function of the dimensionless viscous Stokes layer thickness l+s .The numerical results are compared with the experimental results reported by Tardu4

(symbols in black). The graph shows the incapability of the wall law approach to predictthe experimental wall-shear stress phase shift. The red and blue vertical lines separate theregimes of quasi-laminar and quasi-steady boundary layer behavior. Depending on thepulsation frequency, different boundary layer regimes are experienced. In the quasi-steadyregime, the turbulence has time to relax to the local (in time) equilibrium. The flow canbe studied as a succession of steady states and the wall function assumption seems to bestill valid in this flow conditions as shown in figure 6. When the frequency is increasedthe turbulence production and dissipation start to show a phase lag. In this situation achange in amplitude and phase of the wall-shear stress in respect of the outer velocity ismeasured. A Stokes layer, where the effects of the outer flow oscillations are confined,occurs. The thickness of the Stokes Layer decreases with increasing forcing frequencyand in the inertia dominated or quasi-laminar regime the Stokes layer resides completelywithin the viscous sub layer. In this case a flow solution can be obtained combining the

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laminar Stokes solution in the laminar sub-layer with a turbulent plug flow far from thewall.

5 LES and URANS Near Wall Numerical Predictions in Turbulent Pulsating

Flows

For the comparison between URANS and LES near-wall numerical prediction in turbu-lent pulsating flow we designed a LES pulsating channel case in order to meet the value ofls+ considered in the Tardu et al.4. The computational domain and boundary conditionsare analogous to the validating oscillating case in section 3.2. The source term has beenmodified as follows:

−1

ρ

∂P

∂x= Kosc sin ωt + K (67)

Kosc in first approximation has been evaluated as in section 3.2. K has been chosen,as a first approximation, to balance the wall mean value shear-stress:

K =2τw

h(68)

where h is the channel heigth and in our case the size of the cubic numerical domainchosen. For the case of l+s = 14.14 we considered a value of Uc = 70 m/s and an heigthof the channel h = 0.006 m. We obtained then using steady channel correlation a valueof τw = 13.88 N/m2 and ¯uτ = 3.48 m/s with a channel Reynolds number of 98800. Theconsequent pulsation frequency can be sought from the l+s definition and it is around 1140Hz.

The amplitude of the velocity pulsation (Auc) it has been chosen equal to 20 m/s. Fordifferent value of l+s only the oscillating frequency has been changed in the evaluationof the source terms. All the computations are then made with the same channel size(h = 0.006 m) and the computed values of Uc and Auc are always fixed around 70 and 20m/s.

For the Wall-Function computation an equally spaced grid 31x31x11 has been used.For the Wall-Normal Resolved computation a wall-normal stretched grid 21x21x21 withan hyperbolic tangent stretching law has been employed. For comparison, a DNS channelof h = 0.0015 m with same values of Uc and Auc has also been computed. In order tomatch the nominal condition of l+s = 14.14 the pulsation frequency has been set to 1500Hz. The DNS grid consists of a wall-normal stretched grid with 73x73x9 points.

The results are shown in figure 6

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6 Conclusions

Figure 15: Wall shear stress phase shift dependence on pulsation frequency

The use of wall-functions for LES and URANS has been investigated paying specialattention on the wall shear stress phase shift. The LES computations were validatedusing the DNS results from Spalart and Baldwin3 on oscillating flows showing a goodagreement between the Wall-Function and Wall-Normal Resolved approaches. When thetwo near wall modeling were tested in pulsating conditions discrepancies have been foundin the wall-shear stress phase shift predictions. Similar results have been obtained alsofor URANS calculations using as a test case the experimental results from Tardu et al.4.It is interesting at this stage to point out that the oscillations in the case of Spalart andBaldwin are well above the quasi-steady regime. The l+s parameter in this case has beencomputed using the maximum skin friction velocity during the period of oscillation. We donot expect strong phase shift effects in this case and for this reasons the validation resultsconfirm our findings. The Wall-Normal Resolved computations and the Low-ReynoldsTurbulent model seem to capture the unsteady effects of pulsation the wall-shear stressphase shift. The use of wall functions is than accurate only in cases in which the oscillationare well above the quasi-steady regime. In all the other cases the use of wall-functions inURANS and LES computations is questionable especially in applications in which phaselags can play an important role such as the prediction of thermo-acoustic instabilities.

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REFERENCES

[1] H. G. Weller, G. Tabor, H. Jasak and C. Fureby. A Tensorial Approach to Compu-tational Continuum Mechanics using Object Orientated Techniques. Computers inPhysics, 12-6, 620–631, (1998)

[2] V. Moureau and G. Lartigue and Y. Sommerer and C. Angelberger and O. Colinand T. Poinsot. Numerical Methods for Unsteady Compressible Multi-ComponentReacting Flows on Fixed and Moving Grids Journal of Computational Physics, 202-

2, 710–736, (2005).

[3] P. R. Spalart and B. S. Baldwin. Direct Simulation of a Turbulent Oscillating Bound-ary Layer. Turbulent shear Flows 6, Springer-Verlag, (1989).

[4] S. F. Tardu and G. Binder and R. F. Blackwelder. Turbulent Channel Flow withLarge-Amplitude Velocity Oscillations. Journal of Fluid Mechanics, 267, 109–151,(1994).

[5] J. Smagorinsky. General Circulation Experiments with the Primitive Equations, I.The Basic Experiment. Monthly Weather Review, 91-3, 99–164, (1963).

[6] F. Ducros, F. Nicoud and T. Poinsot. Wall-Adapting Local Eddy-Viscosity Modelsfor Simulations in Complex Geometries. ICFD, 293–300, (1998).

[7] R. I. Issa. Solution of the Implicitly Discretised Fluid flow Equations by Operator-Splitting. Journal of Computational physics, 62, 40–65, (1985).

[8] H. Schlichting and K. Gersten. Boundary Layer Theory, Springer Verlag, (1999).

[9] S. Fan, B. Lakshminarayana and M. Barnett. Low-Reynolds-Number k-ǫ Model forUnsteady Turbulent Boundary-Layer Flows. AIAA Journal, 31-10, (1993).

[10] V. C. Patel, W. Rodi and G. Scheuerer.Turbulence Models for Near-Wall and LowReynolds Number Flows: A Review. AIAA Journal, 23-9, (1985).

[11] B. E. Launder and B. I. Sharma. Application of the Energy Dissipation Model ofTurbulence to the Calculation of Flow Near a Spinning Disc. Letters in Heat andMass Transfer, 1, 131-138, (1974).

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les and urans unsteady boundary layer strategies for pulsating and oscillating...

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