LES of a turbulent jet impinging on a heated wall using high-order numerical schemes

International Journal of Heat and Fluid Flow xxx (2014) xxx–xxx

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LES of a turbulent jet impinging on a heated wall using high-order numerical schemes T. Dairay a,⇑, V. Fortuné b,1, E. Lamballais b, L.E. Brizzi b a

Turbulence, Mixing and Flow Control Group, Department of Aeronautics, Imperial College London, London SW7 2AZ, United Kingdom Institute PPRIME, Department of Fluid Flow, Heat Transfer and Combustion, CNRS – Université de Poitiers ENSMA, Téléport 2, Boulevard Marie et Pierre Curie, BP 30179, 86962 Futuroscope Chasseneuil Cedex, France b

a r t i c l e

i n f o

Article history: Received 19 October 2013 Received in revised form 26 July 2014 Accepted 1 August 2014 Available online xxxx Keywords: Impinging jet Direct numerical simulation Large-eddy simulation Subgrid-scale model Spectral vanishing viscosity Eddy viscosity model

a b s t r a c t Large-eddy simulations (LES) of a turbulent impinging jet flow with a nozzle-to-plate distance of two jet diameters and a Reynolds number of Re ¼ 10; 000 are presented in comparison with experimental data and results from a Direct Numerical Simulation (DNS). The impingement wall is heated and both dynamical and thermal features of the flow are discussed. It is shown that highly accurate numerical methods can lead to correct predictions of velocity statistics and heat transfer but only if a procedure is used to regularize the large-scale dynamics computed explicitly. A better regularization is obtained using a numerical dissipation that mimics a spectral vanishing viscosity in comparison to conventional subgrid-scale models based on an eddy viscosity assumption, even though the model is adapted to the wall through an explicit correction or using the dynamic procedure. These observations suggest that, in the present context of high-order schemes, a simple high-order artificial dissipation coherent with the numerical methods is more suitable than a physical subgrid-scale model that does not take the numerical errors into account. The radial evolution of the Nusselt number predicted by DNS is non-monotonous but this specific behavior is not captured by LES. Due to the complexity of the turbulent processes associated with the corresponding secondary peak in the Nusselt number distribution, an explicit calculation of all the significant scales seems to be required. This conclusion goes against the use of a coarse LES grid in this region of the flow. Ó 2014 Elsevier Inc. All rights reserved.

1. Introduction Impinging jets are widely used in industrial applications as efficient tools to enhance heat transfer between a fluid and an impinged solid target. During the last fifty years, they have been the subject of extensive experimental research summarized in many review papers (e.g. Gauntner et al., 1970; Jambunathan et al., 1992; Webb and Ma, 1995). However the connection between the vortical structures in the flow and the heat transfer distribution at the wall remains unclear. In particular, experimental studies have shown the appearance of a secondary peak in the radial evolution of the mean Nusselt number (e.g. Fénot et al., 2005a,b). The identification of the physical mechanisms responsible for this phenomenon is extremely difficult, since measurement techniques cannot provide the evolution of the instantaneous fields in all space simultaneously. ⇑ Corresponding author. Tel.: +44 2075945078. E-mail addresses: [email protected] (T. Dairay), veronique.fortune@ univ-poitiers.fr (V. Fortuné). 1 Tel.: +33 549454044; fax: +33 549453663.

Concerning the numerical studies, Reynolds Average Navier– Stokes (RANS) models have shown their limits to predict the impinging jet flow: (i) erroneous prediction of the stagnation zone flow, (ii) overprediction of the stagnation point heat transfer, (iii) large number of arbitrary coefficients leading to poor reproducibility of the models (Dewan et al., 2012; Zuckerman and Lior, 2005). In the last decade, unsteady numerical simulations (LES/DNS) of impinging jet flow have been carried out. For the turbulent case, LES is mainly used (e.g. Hadziabdic and Hanjalic, 2008; Uddin et al., 2013) but some DNS results at moderate Reynolds number (typically Re 6 2000) are also documented (Chung et al., 2002; Tsubokura et al., 2003; Rohlfs et al., 2012). In the present study, both DNS and LES are used to compute a fully turbulent impinging jet flow with a nozzle to plate distance of H ¼ 2D and a Reynolds number of Re ¼ 10; 000. For the DNS, a very fine grid is used in order to reach a Reynolds number that has never been considered before to the authors knowledge. These results are validated by comparison to experimental measurements. Using the DNS database, the main goal of the study is to compare two types of LES strategies based either on eddy viscosity

http://dx.doi.org/10.1016/j.ijheatfluidflow.2014.08.001 0142-727X/Ó 2014 Elsevier Inc. All rights reserved.

Please cite this article in press as: Dairay, T., et al. LES of a turbulent jet impinging on a heated wall using high-order numerical schemes. Int. J. Heat Fluid Flow (2014), http://dx.doi.org/10.1016/j.ijheatfluidflow.2014.08.001

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assumption or on high-order numerical dissipation to mimic the subgrid-scale contribution. The actual role of subgrid-scale modeling is assessed with reference to a counterpart LES free from any subgrid-scale model. In terms of methodology, the main originality of this work is that the LES assessment is carried out in the context of highly accurate numerical methods. The use of high-order schemes leads to specific difficulties that must be taken into account for a reliable production of LES results. High-order schemes are very attractive for DNS due to their ability to provide accurate results using a moderate number of degrees of freedom, a property that allows a high computational efficiency. However, in the context of LES, the sensitivity of highorder schemes at small scale can be counterproductive if there is a direct source of numerical errors at scales close to the mesh size. As a main source of errors, aliasing must be mentioned (Kravchenko and Moin, 1997), especially if the truncation associated with the computational mesh corresponds to a wave number where the turbulent kinetic energy is still significant in the spectrum. Paradoxically, a subgrid-scale model that is designed (at least in part) to ensure subgrid-scale dissipation, can also be a strong source of numerical errors. For instance, for a model based on eddy viscosity assumption, the extra non-linearity introduced in the LES equations produces additional aliasing errors. These errors could be controlled using a mesh size significantly thiner than the LES filter, but due to the corresponding drastic increase of the computational cost, this strategy is almost never adopted. When low-order (typically second-order) schemes are used, aliasing errors are often negligible, meaning that numerical errors associated with the discretization of the subgrid-scale model can be accepted. This is not the case of high-order methods, and this specific problem is investigated in this paper. Here, an original way to control small-scale numerical errors while mimicing the subgrid-scale contribution is proposed as an alternative to conventional subgrid-scale models. The idea is to make coherent the regularization procedure with the numerical methods used to solve the governing equations. The point is to determine how far this strategy can be beneficial with respect to conventional subgrid-scale modeling. The paper is organized as follows: Section 2 presents the flow configuration and the numerical methods including boundary conditions. Then Section 3 gives details about the original way of using numerical dissipation as a subgrid-scale model while reminding the main properties of the explicit eddy viscosity and diffusivity models used in this study. After a brief presentation of the simulation parameters in Section 4, a set of results is presented in Section 5 to investigate the ability of LES to reproduce both velocity and temperature statistics in comparison to experimental results and DNS. Instantaneous flow vizualisations are also used to highlight the crucial role of small-scale turbulence in the near-wall dynamics and heat transfer. The main conclusions of the study are provided in Section 6. 2. Physical and numerical parameters 2.1. Flow configuration In the Cartesian coordinate system ðO; x; y; zÞ, the computational   domain is X ¼ ½Lx =2; Lx =2  0; Ly  ½Lz =2; Lz =2 where the origin O is located at the center of the plate (see Fig. 1 for a schematic pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi view). For the sake of simplicity the radial distance r ¼ x2 þ z2 and the azimuthal angle h ¼ arctanðx=zÞ are also introduced hereinafter. The Reynolds number of the flow is based on the jet diameter D and its (constant) bulk velocity U b with

Re ¼

Ub D

m

¼ 10; 000

ð1Þ

Fig. 1. Schematic representation of the computational domain.

where m is the (constant) kinematic viscosity. The Prandtl number is set to

Pr ¼

m ¼1 j

ð2Þ

where j is the (constant) thermal diffusivity of the fluid. The nozzle-to-plate distance H corresponds to the computational domain height Ly , with H=D ¼ 2. For this particular ratio, a wellmarked secondary peak is expected for the mean Nusselt number (see Section 5). 2.2. Numerical methods The governing equations are the incompressible Navier–Stokes and temperature equations    @u 1 1 þ ðr  ðu  uÞ þ ðu  rÞuÞ ¼  rp þ r  ðm þ mt Þ ru þ ruT ð3Þ @t 2 q ru¼0 ð4Þ @T þ u  rT ¼ r  ½ðj þ jt ÞrT  ð5Þ @t  T where u ¼ ux ; uy ; uz is the velocity, p the pressure and T the temperature. In the present framework, the density q is constant so that the temperature behaves like a passive scalar. For DNS, mt ¼ jt ¼ 0 whereas for LES, Eqs. (3)–(5) are the filtered equations for which unknowns are filtered quantities while p is a modified pressure including a subgrid-scale contribution (Sagaut, 2006; Lesieur et al., 2005). The presence of the eddy viscosity mt and diffusivity jt indicates that when an explicit subgrid-scale modeling is used, it is based on the Boussinesq hypothesis. Note that the alternative subgrid-scale modeling proposed in this work is included in the viscous terms of Eqs. (3) and (5) so that as in DNS, we have mt ¼ jt ¼ 0 for this type of LES (see Section 3.1). Eqs. (3)–(5) are numerically solved on a Cartesian mesh using a code called ‘‘Incompact3d’’. This solver is based on sixth-order centered compact finite difference schemes (Lele, 1992) for the spatial discretization and an hybrid explicit/implicit third-order Adams– Bashforth/second-order Crank–Nicolson scheme for the time advancement. The pressure mesh is staggered from the velocity mesh to avoid spurious pressure oscillations. With the help of the modified wave number concept, the divergence free condition is ensured up to machine accuracy. More details about the present code and its validation, especially the original treatment of the pressure in the spectral space, can be found in Laizet and Lamballais (2009) and Laizet et al. (2010). Here, the massively parallel version of the code is used through an MPI implementation

Please cite this article in press as: Dairay, T., et al. LES of a turbulent jet impinging on a heated wall using high-order numerical schemes. Int. J. Heat Fluid Flow (2014), http://dx.doi.org/10.1016/j.ijheatfluidflow.2014.08.001

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based on pencil domain decomposition strategy (see Laizet and Li, 2011, for more details). 2.3. Boundary conditions at the plates No-slip boundary conditions are applied to the impingement plate y ¼ 0 and to the confinement plate y ¼ Ly for r P D=2. To model the jet at the nozzle exit for r < D=2 at y ¼ Ly , a mean velocity profile huy i is prescribed as inflow boundary conditions

huy i ¼ 

  n  nþ2 2r Ub 1 n D

ð6Þ

u0 ¼ f ðrÞ

N X

Am ðtÞ cos ðmh þ /m ðtÞÞ

ð7Þ

m¼1

for each velocity component where N is the number of excited azimuthal modes and Am and /m are amplitude and phase generated randomly up to a cutoff frequency. The use of a limited number of azimuthal modes up to a moderate cutoff frequency avoids spurious excitations on space and time scales that cannot be accurately captured by the mesh. The modulation function f ðrÞ in Eq. (7) is adjusted in shape and amplitude to match roughly the experimental conditions that were used as reference for validation (see Section 5). For the temperature field, a constant flux density up < 0 is prescribed on the heated impingement plate (y ¼ 0) with

up ¼

@T @y y¼0

ð8Þ

See Fig. 1 for a schematic view. The prescription of non-homogeneous Neumann boundary conditions using compact finite difference schemes is straightforward when the time integration is fully explicit (e.g. Slinn, 1995). However, keeping the compact structure of the schemes is more difficult and less computationally efficient when implicit time integration is used (e.g. Zhao et al., 2007). For the Crank–Nicolson scheme used here in the axial direction y, the prescription of a constant heat flux on the impingement plate is achieved in a simple way using an explicit third-order finite difference scheme at the boundary nodes with

up ¼

11T 1 þ 18T 2  9T 3 þ 2T 4 6 Dy

2.4. Outflow boundary conditions Near the lateral faces of the domain (i.e. x=D ¼ Lx =2 and z=D ¼ Lz =2), the fringe method proposed by Nordstrom et al. (1999) is used. This technique consists in the addition of a volume force F in the right-hand side of the Navier–Stokes Eq. (3) with

  e u F ¼ kðrÞ u

ð9Þ

where T j ¼ Tðyj Þ refers to the values of the temperature field at the nodes yj ¼ ðj  1ÞDy and Dy is the mesh spacing (a regular mesh is assumed for clarity). In the implicit time advancement procedure, scheme (9) is used to provide locally the wall temperature leading to the expected flux density up . Concerning the last node, the isothermal condition T ¼ T j on the confinement plate or at the jet inflow (y ¼ Ly ) is prescribed as a Dirichlet boundary condition. This combination leads to a narrow band implicit matrix with 9 diagonals. Apart from this particular Neumann boundary condition combined with the time implicit treatment of the second derivative in the y-direction, the rest of numerical methods are identical to those reported in Laizet and Lamballais (2009).

ð10Þ

e is a target velocity field and kðrÞ a modulation function where u allowing a local activation of the forcing in the region where k – 0. Here, the corresponding fringe region is defined using

kðrÞ ¼

where the exponent n can be chosen to mimic a more or less flat velocity profile. The correction ðn þ 2Þ=n enables the bulk velocity to be equal to U b for any value of n. The same synthetic profile was used by Tsubokura et al. (2003) with n ¼ 8. Here, to mimic a top-hat velocity profile associated with a short convergent nozzle, which is well known to lead to a non-developed flow at the exit, a higher value for n was used with n ¼ 28. This particular geometry corresponds to the most typical situation in industrial applications. The jet is excited using synthetic perturbations of the form

3

1 ð1 þ tanh ðbðr  r m Þ  4DÞÞ 2

ð11Þ

where r m ¼ 5D corresponds to the minimal radius for which the forcing is activated while b is a parameter controlling the spatial stiffness of this activation. As target velocity field, a purely radial flow is assumed with

e¼ u

 2 ! 4 y  Ly =2 3 U d D2 er 1 16 Ly r L2y

ð12Þ

This artificial flow can be seen as a Poiseuille flow ponderated with 1=r in order to ensure mass conservation and then avoid any conflict with the divergence free condition (4). Practically, this forcing damps very efficiently the flow perturbations in the neighborhood of the minimal radius r P rm while allowing a radial evacuation of the flow through the lateral boundaries without detectable spurious effects. Near the outlet faces, the velocity field is so close to the target velocity than the latter can be used as a simple Dirichlet outflow boundary condition. Preliminary tests (reported in Dairay et al., 2011) have shown that for the present flow configuration, the use of this fringe method is highly preferable to a conventional convective outflow boundary condition as described for example by Ol’shanskii and Staroverov (2000). Comparing these two types of outflow boundary conditions in 2D simulation of plane impinging jet, Dairay et al. (2011) have observed that the use of a convection equation as outflow boundary conditions leads to a specious excitation of the jet in the inflow region. This behavior is due to a spurious interaction between the inlet and the outlet. The numerical errors on the outflow boundary conditions are transmitted to the inflow through the pressure field used to ensure incompressibility. This numerical artifact, observed for instance in a spatially developing mixing layer (Buell and Huerre, 1988), can be very misleading for the present impinging jet flow where outflow boundary conditions must be treated up to 4 faces of the computational domain (in the 3D case). To illustrate this phenomenon, the instantaneous vorticity fields of two 2D simulations obtained at the same time are plotted in Fig. 2. For the first simulation, the fringe method is used whereas for the second one, a convective outflow boundary condition is used. A drastic difference can be observed in the dynamics, with a more stable and realistic development of the jet using the fringe method. A particularly misleading feature is that both simulations could be seen as physically acceptable if only one snapshot was considered. However, a temporal analysis of the flow shows that when convective outflow boundary conditions are used, the jet seems to be highly self-excited with an unrealistic flapping of high amplitude even if the jet is not perturbated at the inflow (see Fig. 2-bottom). On the contrary, under the same conditions, the use of the fringe method allows to generate a quasi-steady jet free from any spurious flapping as expected (see Fig. 2-top). The use of the fringe method is based on the assumption that the physics of the flow is not modified outside the neighborhood of the forcing region. To check this point, additional 2D and 3D calculations have been performed while considering more or less

Please cite this article in press as: Dairay, T., et al. LES of a turbulent jet impinging on a heated wall using high-order numerical schemes. Int. J. Heat Fluid Flow (2014), http://dx.doi.org/10.1016/j.ijheatfluidflow.2014.08.001

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Fig. 2. Instantaneous vorticity field obtained with two 2D simulations at the same time. Top: using a buffer zone and bottom: using a convective outflow boundary condition.

extended computational domain and buffer zones. Our conclusion was that when the forcing is applied beyond a critical radius of about r c ¼ 5D, no significant influence was observed for r < rc  D. Even the exact form of the target solution (12) was found to be of secondary importance providing that good damping of perturbations can be ensured near the outflow boundaries. Concerning the temperature, an outflow boundary condition is provided by solving the purely convective equation

on the concept of Spectral Vanishing Viscosity (SVV, see for instance Tadmor, 1989; Karamanos and Karniadakis, 2000; Pasquetti, 2006). In this study, a sixth-order accurate version of this SVV-like operator is proposed. The principle of the method is exactly the same as in Lamballais et al. (2011) except that an extra coefficient is introduced, leading to the following 3–9 stencil formulation

@T þ u  rT ¼ 0 @t

af 00i1 þ f 00i þ af 00iþ1 ¼ a

ð13Þ

Note that because this equation is solved at the outlet, where the forcing of the fringe method is very active, the velocity vector u is virtually equal to its target value given by (12), with the corresponding flow being stationary, purely radial and axisymmetric. This conventional outlet treatment has been found fully satisfactory (no significant artefact near the lateral boundaries) while avoiding the difficulty of prescribing a reliable target temperature field as required by the fringe method. 3. Subgrid-scale modeling and regularization 3.1. High-order numerical dissipation for DNS and LES Recently, Lamballais et al. (2011) suggested a method enabling the introduction of a controlled numerical dissipation restricted to

00

k Dx2 ¼

f iþ1  2f i þ f i1 f  2f i þ f i2 þ b iþ2 Dx2 4 Dx2 f  2f i þ f i3 f  2f i þ f i4 þ c iþ3 þ d iþ4 9 Dx2 16Dx2 00

00

where f i ¼ f ðxi Þ and f i ¼ f ðxi Þ denote the values of the function f ðxÞ 00 and its second derivatives f ðxÞ at the nodes xi ¼ ði  1ÞDx with Dx the uniform mesh spacing. For the present 5 parameters ða; a; b; c; dÞ scheme, the choice is to preserve only 3 constraints: (i) a þ b þ c þ d ¼ 1 þ 2a (Dx2 condition); (ii) a þ 4b þ 9c þ 16d ¼ 12a (Dx4 condition); (iii) a þ 16b þ 81c þ 256d ¼ 30a (Dx6 condition). This choice provides the sixth-order accuracy while leaving free 2 coefficients. In the framework of Fourier analysis, it is well known that a 00 modified square wavenumber k can be related to the scheme (14) through the expression

2a½1  cosðkDxÞ þ 2b ½1  cosð2kDxÞ þ 2c ½1  cosð3kDxÞ þ 8d ½1  cosð4kDxÞ 9 1 þ 2a cosðkDxÞ

a selected range of scales close to the mesh size. This method is based on specific high-order finite difference schemes for the computation of the second derivatives in the viscous terms. The schemes are centered so that no upwind treatment is required contrary to the popular use of non-centered dissipative schemes for the computation of first derivative in the convective terms. The dissipation can be easily controlled through the coefficients of the scheme while preserving its formal accuracy. In Lamballais et al.’s (2011) study, two sets of scheme coefficients are distinguished. The first coefficient set leads to a sixth-order scheme that introduces a small amount of numerical dissipation only at the smallest scales resolved by the mesh, in a spectral range where numerical errors are inevitably high due, for instance, to aliasing or dispersion errors. This scheme, wellsuited for DNS, has been used for the DNS presented in this study. A second set of coefficients is provided by Lamballais et al. (2011) to better control the numerical dissipation while extending its influence on a wider range of scales. The resulting fourth-order scheme is designed to mimic a subgrid-scale model for LES based

ð14Þ

ð15Þ

where k is the actual wave number. Then, two additional relation00 ships can be imposed through a condition on k at the cutoff 00 00 wavenumber kc ¼ p=Dx with k p ¼ kc and another one at the 00 00 intermediate scale 2kc =3 with k 2p=3 ¼ km . It is easy to show that the relationships (i, ii, and iii) combined with these two requirements lead to the set of coefficients 00 320km Dx2  1296 00 00 2 405kc Dx  640km Dx2 þ 144 00 00 00 00 2 4329kc Dx =8  32km Dx2  140kc Dx2 km Dx2 þ 286 a¼ 00 00 405kc Dx2  640km Dx2 þ 144 00 00 00 00 2115kc Dx2  1792km Dx2  280kc Dx2 km Dx2 þ 1328 b¼ 00 00 405kc Dx2  640km Dx2 þ 144 00 00 00 00 7695kc Dx2 =8 þ 288km Dx2  180kc Dx2 km Dx2  2574 c¼ 00 00 2 2 405kc Dx  640km Dx þ 144 00 00 00 00 198kc Dx2 þ 128km Dx2  40kc Dx2 km Dx2  736 d¼ 00 00 405kc Dx2  640km Dx2 þ 144

1 2

a¼ 

ð16Þ

Please cite this article in press as: Dairay, T., et al. LES of a turbulent jet impinging on a heated wall using high-order numerical schemes. Int. J. Heat Fluid Flow (2014), http://dx.doi.org/10.1016/j.ijheatfluidflow.2014.08.001

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The extra-dissipation introduced by the discrete viscous operator can be interpreted as a spectral viscosity expressed as 2

00

m00s ¼ m

k k

ð17Þ

2

k

See Lamballais et al., 2011, for more details. Using this expression, it is easy to adjust the set of coefficients (16) in order to mimic roughly the SVV kernel that reads

 2 ! kc  k ms ðkÞ ¼ m0 exp  0:3kc  k

ð18Þ

where kc is the cutoff wave number of the computational grid and

m0 an artificial viscosity that controls the numerical extra-dissipation. Here, as in Lamballais et al. (2011), only the two conditions m00s ðkc Þ ¼ ms ðkc Þ and m00s ð2kc =3Þ ¼ ms ð2kc =3Þ are imposed. The ability of the present finite difference scheme to behave like a SVV operator is illustrated in Fig. 3 with the value m0 ¼ ms ðkc Þ actually used in the LES based on SVV presented in this paper. The spectral viscosity introduced by the sixth-order scheme designed for DNS is also plotted in the same figure to confirm its low-dissipative effect with virtually no dissipation at the scales correctly described by the computational mesh (i.e., at k < 2kc =3). To summarize, two discrete viscous operators are available: one is designed for DNS (O6DNS) and another one is calibrated for LES (O6SVV) with an ability to mimic the SVV kernel. In both operators, a spectral viscosity is embedded, with a vanishing behavior at large scale and no extra computational cost. The small amount of numerical dissipation of the DNS operator is designed to control the numerical errors only at small scale whereas the LES operator can be viewed as a subgrid-scale model. 3.2. Eddy viscosity and diffusivity model Preliminary tests based on the Smagorinsky model

 1=2 mt ¼ C 2s D2 2Sij Sij

ð19Þ

has been performed with the constant C s ¼ 0:065  and the filter size  i =@xj þ @ u  j =@xi =2 is the strain rate D ¼ ðDx Dy DzÞ1=3 where Sij ¼ @ u tensor of the resolved field. It was observed that the resulting incorrect near-wall behavior (overestimation of the subgrid viscosity) prevents to obtain reliable results (expected results not shown for conciseness). It is well known that a clear improvement of the near-wall behavior of the Smagorinsky model can be obtained when a dynamic procedure is applied for the estimation of its ‘‘constant’’

C s (Germano et al., 1991; Lilly, 1992). Here, the localized version of the dynamic Smagorinsky model proposed by Piomelli and Liu (1995) has been used. The test filter is applied in the 3 spatial directions using the sixth-order compact operator proposed by Lele (1992) (set of coefficients C.2.8 page 40) that corresponds to a ratio b and the grid filter width D, i.e. of 2 between the test filter width D b D ¼ 2D. In addition to the localization procedure of Piomelli and Liu (1995), an azimuthal averaging is applied on the constant C s . This averaging is against the generalization of the model but enables a better control of numerical instabilities through the use of a smoother spatially varying C s . For the following comparison, where the ability of models to deal with spurious oscillations will be discussed, this more regular version of the dynamic Smagorinsky model was preferred despite the lack of generality. The Wall Adapting Local Eddy-viscosity model (WALE, see Nicoud and Ducros, 1999, for the reference publication) has also been used in this work to model the subgrid-scale contribution. This model, specifically designed to reproduce the correct wallscaling mt  y3 , computes the eddy-viscosity mt of Eq. (3) from the velocity gradient tensor’s invariants resorting to the following relation

mt ¼

C 2w D2

3=2 sdij sdij

5=2

5=4 Sij Sij þ sdij sdij

ð20Þ

The term sdij is the traceless symmetric part of the square of the  i =@xj , namely resolved velocity gradient tensor gij ¼ @ u

sdij ¼

 1 1 2 g þ g2ji  dij g2kk 2 ij 3

ð21Þ

with g2ij ¼ gik gkj . The model constant is set to C w ¼ 0:5 to perform the present simulation. In the original paper of Nicoud and Ducros (1999), this particular value has been found to give the best results for isotropic turbulence and turbulent pipe flow simulations and it has also been used by Lodato et al. (2009) to perform LES of an impinging jet flow. The subgrid-scale diffusivity is estimated by assuming a constant turbulent Prandtl number Pr t ¼ mt =jt with Pr t ¼ 0:5 for both the dynamic Smagorinsky and WALE models. Note that there is not need of turbulent Prandtl number for the O6SVV model presented in the previous subsection. This can be viewed as an advantage compared to standard eddy viscosity and diffusivity models for which the choice of the value of Prt is difficult to make rigorously. 4. Simulations parameters

25

SVV O6SVV O6DNS

Five calculations of an impinging jet flow at Re ¼ 10; 000 and H=D ¼ 2 are presented. Every simulation uses the same computational domain (see Fig. 1) with Lx ¼ Lz ¼ 12D and Ly ¼ 2D. The Cartesian mesh is regular in the transerve directions and stretched in the axial direction of the jet in order to concentrate the grid points near the impingement plate. The main numerical parameters are reported in Table 1.

20

νs /ν

15

10 Table 1 Summary of the different computational cases. See Fig. 3 for an illustration of the kernel used for the O6DNS and O6SVV second derivative schemes.

5

0 0.001

0.01

0.1

1

k/kc Fig. 3. Spectral viscosities m00s =m used in the calculations reported in Table 1. The SVV kernel is given as reference.

Cases

nx  ny  nz

@ 2 =@x2i schemes

Subgrid-scale model

DNS SVV DSM WALE LDNS

1541  401  1541 257  401  257 257  401  257 257  401  257 257  401  257

O6DNS O6SVV O6DNS O6DNS O6DNS

No model Spectral vanishing viscosity Dynamic Smagorinsky WALE No model

Please cite this article in press as: Dairay, T., et al. LES of a turbulent jet impinging on a heated wall using high-order numerical schemes. Int. J. Heat Fluid Flow (2014), http://dx.doi.org/10.1016/j.ijheatfluidflow.2014.08.001

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The reference calculation, referred as DNS, is a well-resolved DNS that is just slightly regularized using the sixth-order viscous operator described in the previous section (see Fig. 3 for an illustration of the O6DNS kernel). Expressed in wall units at the radial location where friction velocity is maximum, the spatial resolution allows to check Dxþ ¼ Dzþ 10 and 0:9 / Dyþ / 40. The second calculation, referred as SVV, is a LES based on SVV via the O6SVV operator as illustrated in Fig. 3. In comparison with the DNS, the spatial resolution is significantly reduced, leading to Dxþ ¼ Dzþ 55 and 2:5 / Dyþ / 20 in wall units. The third and fourth calculations are also LES with the same spatial resolution as SVV but with a more conventional subgridscale model based on the Boussinesq eddy viscosity hypothesis. Both the DSM and the WALE calculations (see subSection 3.2 for details) were found to lead to correct predictions of the near-wall dynamics. To slightly improve the work of these two models, the sixth-order viscous operator O6DNS is also used as in the DNS because an eddy viscosity model tends to produce spurious oscillations at small scale, especially when high-order methods are used. The O6DNS kernel (see Fig. 3) was found to reduce slightly these oscillations without introducing significant numerical dissipation in comparison with the dissipation of the subgridscale model. The fifth calculation, referred as LDNS, uses the same spatial resolution as SVV, DSM and WALE but without any attempt to model the subgrid-scale contribution. This simulation can be viewed as a LES without any subgrid-scale model or as a ‘‘false’’ DNS with a too coarse mesh regarding the small-scale physics involved at this Reynolds number. The same sixth-order dissipation as in DNS, DSM and WALE is used to ensure the minimal regularization of the solution at small scales, but with a resulting negligible dissipation compared to what is required to represent the subgrid-scale contribution. In the following, statistical and instantaneous results are compared for the five calculations. The DNS case is considered as the reference case in the sense that the computational configuration is the same as for the four other calculations (same computational domain, boundary conditions, numerical methods, etc.) except for the spatial resolution that is significantly higher. For validation purpose, results are also discussed in comparison with their experimental counterparts obtained in the flow configuration which the present simulations have been calibrated on (same Reynolds number and similar inflow conditions). For details concerning the experimental setup used for velocity measurements the reader may refer to the papers of Roux et al. (2011) and Dairay et al. (2013). 5. Results For the DNS case, the collection of results for the turbulent statistics is done over a period of 12 cycles, where the cycle period is evaluated from the estimated natural Strouhal number (based on U b and D) St 0:4 corresponding to the impinging frequency of the main large-scale structures in the jet. For the four other cases, that are clearly less computationally expensive, statistics are computed over a longer period of 20 cycles. To improve the statistical convergence, mean fields are averaged both in time and in the homogeneous azimuthal direction. In a cylindrical coordinate system, in the absence of any swirl in the jet, the resulting mean velocity has two non-zero components hur iðr; yÞ and huy iðr; yÞ in the radial and vertical directions respectively. At the wall, the efficiency of the heat transfer can be measured through the instantaneous Nusselt number defined as

Nuðr; h; tÞ ¼

up D Tw  Tj

ð22Þ

where T w ðr; h; tÞ is the instantaneous wall temperature at the impingement plate. The mean Nusselt number hNuiðrÞ is defined in the same way as

hNuiðrÞ ¼

up D hT w i  T j

ð23Þ

where hT w iðrÞ is the mean wall temperature at the impingement plate. 5.1. Velocity statistics For the comparison of dynamical fields statistics, maps of mean velocity magnitude and turbulent kinetic energy are plotted in Fig. 4 for each calculation and for the reference experiment. At first analysis, the classical mean velocity distribution with the free jet (0:5 < y=D < 2), the stagnation (0 < y=D < 0:5 and r=D < 1:8) and the wall jet regions (r=D > 1:8) is well-recovered by all the simulations (see Fig. 4 left column). However, a careful examination of the turbulent kinetic energy maps shows that the LDNS case is not in good agreement with the reference calculation DNS (see Fig. 4 right column). In particular, the near-wall turbulent fluctuations in the impingement region (0 < y=D < 0:2 and r=D < 1) are clearly overpredicted. This overprediction is followed by an underestimation of the near-wall turbulent intensity for r=D > 1 and no wall jet thickening is observed for the LDNS case. A clear improvement of the turbulent kinetic energy map can be observed for the LES SVV, DSM and WALE (see Fig. 4 right column) by reference with the DNS calculation or the measurements. The near-wall turbulence intensity and the wall jet region thickening are well recovered by the three LES despite the underestimation of the turbulence levels in the jet region (1:5 < y=D < 2) from the SVV computation. This improvement of the results is a first indication that a subgrid-scale is useful. These observations are confirmed by the radial velocity profiles plotted in Fig. 5 for the different cases. If the mean radial velocity seems well-predicted by the five computations, it is clear that the near-wall fluctuations are underestimated by the LDNS case in comparison with the three LES cases SVV, DSM and WALE. Note that the high fluctuations levels measured for y=D > 0:6 correspond to erroneous residual fluctuations which are directly linked with the PIV measurement technique and highlighted in the low velocity regions. Despite the use of highly accurate schemes with a small-scale regularization, the lack of any subgrid-scale model in the LDNS case is found unable to provide reliable turbulent fluctuations even if the mean flow seems to be reasonably well-predicted. The main discrepancy in this ‘‘false’’ DNS using a LES resolution is found in the near-wall dynamics, especially in the vicinity of the impingement region. 5.2. Temperature statistics The radial distribution of the mean Nusselt number is plotted in Fig. 6-left for each calculation. The Nusselt number distribution of the reference DNS computation shows the classical behavior of a turbulent impinging jet heat transfer at small nozzle-to-plate distance (Lee and Lee, 1999) with a primary maximum for r=D 0:7 and a secondary maximum for r=D 2 (see Fig. 6). For the LES with or without subgrid model, contrasting results are obtained. For the LDNS calculation, the Nusselt number is strongly overpredicted in the impingement region (r=D < 1). This overprediction can be related with the erroneous generation of turbulent fluctuations in the same region as observed in the previous subsection. The discrepancy of NuðrÞ with respect to the DNS is clearly attenuated for the DSM and WALE computations, but a clear anomalous behavior of the heat transfer can be noticed near the stagnation point.

Please cite this article in press as: Dairay, T., et al. LES of a turbulent jet impinging on a heated wall using high-order numerical schemes. Int. J. Heat Fluid Flow (2014), http://dx.doi.org/10.1016/j.ijheatfluidflow.2014.08.001

T. Dairay et al. / International Journal of Heat and Fluid Flow xxx (2014) xxx–xxx

7

Fig. 4. Maps of the mean velocity magnitude and turbulent kinetic energy obtained from the experimental data of Dairay et al. (2013) and from the five present calculations.

The best Nusselt prediction is provided by the SVV calculation that is able to capture very accurately the first peak at r=D 0:7 in agreement with the DNS. At larger radius, the agreement is less favorable: LES misses the local minimum of NuðrÞ at r=D 1:4 so that no well-marked secondary peak can be correctly captured. Only a slight slope change can be noticed for the cases WALE and SVV in the secondary peak region. The observation of the fluctuating wall temperature standard deviations plotted in Fig. 6-right confirms the erroneous behavior of the LDNS, DSM and WALE cases in the impingement region (r=D < 1). Regarding the temperature statistics, it can be concluded that a subgrid-scale model is crucial to capture realistically the heat transfer in the impingement region. For the present code based

on high-order schemes, the SVV approach is the most accurate in terms of thermal prediction without any anomalous feature in the stagnation region unlike the behavior observed using the dynamic Smagorinsky or WALE models. However, in a more distant region from the impingement, the radial distribution of the mean Nusselt number is not accurately predicted with an almost monotonic decrease in contrast to the DNS results. The inability of LES to capture the secondary peak of the Nusselt number can be attributed to a modeling failure or to specific small-scale turbulent structures that cannot be explicitly calculated while being very difficult to model as a subgrid-scale contribution. This point will be discussed in the next section by investigation of the 3D unsteady turbulent structures actually computed by DNS and LES.

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EXPE DNS SVV WALE DSM LDNS

1.6 1.4 1.2

1.4 1.2 1

y/D

1

y/D

EXPE DNS SVV WALE DSM LDNS

1.6

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0 -0.1

0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0

0.05

0.1

0.15

ur

0.2

0.25

ur2

Fig. 5. Mean radial velocity (left) and fluctuating radial velocity standard deviation (right) at the location r=D ¼ 2 obtained from the experimental data of Dairay et al. (2013) and from the five present calculations.

110

0.006

DNS SVV WALE DSM LDNS

100 90

DNS SVV WALE DSM LDNS

0.005 0.004

Nu

Tw2

80

0.003

70 0.002

60

0.001

50 40

0 0

0.5

1

1.5

2

2.5

3

r/D

0

0.5

1

1.5

2

2.5

r/D

Fig. 6. Radial distribution of the mean Nusselt number (right) and fluctuating wall temperature standard deviation (right) on the impingement plate.

DNS

SVV

DSM

LDNS

WALE

Fig. 7. Instantaneous maps of the Nusselt number on the impingement plate obtained from the five present computations.

Please cite this article in press as: Dairay, T., et al. LES of a turbulent jet impinging on a heated wall using high-order numerical schemes. Int. J. Heat Fluid Flow (2014), http://dx.doi.org/10.1016/j.ijheatfluidflow.2014.08.001

T. Dairay et al. / International Journal of Heat and Fluid Flow xxx (2014) xxx–xxx

DNS

SVV

DSM

LDNS

9

WALE

Fig. 8. Instantaneous isosurface 30U 2b =D2 of Q criterion obtained with the five present computations. The isosurface is colored with the axial distance from the impingement plate y=D with a color map saturated for y=D > 0:5. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

5.3. Instantaneous flow vizualisations In order to analyze the unsteady processes associated with the heat transfer at the wall, instantaneous Nusselt number maps are plotted in Fig. 7 for each calculation. The pattern obtained by the SVV calculation is clearly in better qualitative agreement with the DNS reference case in comparison with the cases LDNS, DSM and WALE. This is particularly true in the impingement region, but also at larger radial distance where the numerous ‘‘cold spots’’ (regions of high values of the Nusselt number) observed in DNS are wellrecovered in the SVV case both in terms of scale and shape. At the same location, these spots are more fragmented and artificially stretched in the radial direction for the LDNS, DSM and WALE cases, while clearly exhibiting the presence of spurious numerical oscillations. Because the instantaneous Nusselt number is the footprint of dynamical processes in the near-wall region, it can be instructive to visualize instantaneous turbulent structures in this zone. Here, the Q-criterion is used as vortex identification technique (Hunt et al., 1988; Dubief and Delcayre, 2000) with

Q ¼

@uj @ui @xi @xj

ð24Þ

A characteristic isosurface of this quantity, identical for all the calculations, is plotted in Fig. 8. For the reference DNS case, an impressive number of various turbulent structures is detected. These structures are highly 3D in a wide range of scales even if a background large-scale organization is also clearly visible in both the perspective and top views (see Fig. 8 top left). In the impingement region, a periodic formation of rolls with a strong azimuthal coherence is observed at a frequency St 0:4. These rolls are distorted by the numerous small-scale structures stretched in the radial direction. As expected, due to the coarse grid used in LES, these small-scale vortices are not present in the SVV calculation (see Fig. 8 top middle) but the large-scale azimuthal structures seem to be correctly captured with a passing frequency in very good agreement with the DNS. Qualitatively, a natural filtering is applied to the solution that seems to be free from numerical artifacts. The same cannot be said for the LDNS (see Fig. 8 bottom middle), DSM (see Fig. 8 bottom left) and WALE (see Fig. 8 top right) cases for which the present vortex identification is strongly interfered by numerical errors. These cases reveal the presence of spurious numerical oscillations that the subgrid-scale models DSM and WALE (combined with the O6DNS numerical dissipation) are unable to control. These unrealistic

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visualizations, based on the Q-criterion that involves velocity derivatives, can explain the erroneous pattern of the instantaneous Nusselt number as well as the resulting error concerning the averaged heat transfer in the impingement region.

6. Conclusion One DNS and four LES of a turbulent impinging jet flow have been carried out using high-order numerical methods and a careful prescription of boundary conditions. The DNS results are used as reference to compare two strategies of LES modeling, a first one based on a conventional eddy viscosity assumption and a second one based on high-order numerical dissipation to mimic a spectral vanishing viscosity. Both types of LES are found to lead to acceptable velocity statistics in comparison with DNS and experimental results of reference. In particular, in comparison with a LES free from any subgrid-scale modeling (or equivalently to a ‘‘false’’ DNS based on a too coarse mesh), the use of subgrid-scale model is found highly beneficial to obtain reliable turbulent kinetic energy. The analysis of temperature statistics confirms these observations with an additional conclusion. A conventional subgrid-scale model based on eddy viscosity (here, the dynamic Smagorinsky and WALE models) is found unable to control the numerical errors at small scale, leading to unrealistic heat transfer predictions in the impingement region. This behavior is interpreted as a consequence of the aliasing errors not controlled by the model when high-order numerical schemes are used as in the present study. Due to presence of strong nonlinearities in their expression, the dynamic Smagorinsky and WALE models are likely to be themselves a significant source of aliasing errors. In this particular context, the use of a discrete viscous operator that mimics a spectral vanishing viscosity seems to be a preferable option. This regularization technique efficiently damps spurious small-scale oscillations. This feature is beneficial in terms of physical realism, in particular for the faithful description of the large-scale turbulent structures that can be accurately captured by the LES mesh. More quantitatively, this ability is found to improve significantly the heat transfer statistics at the wall, especially in the impingement region. The present impinging jet configuration is known to be highly challenging due to the occurence of a secondary peak in the Nusselt number distribution. The physical reasons of this phenomenon are difficult to analyze and require more investigation. As a first indication, it is interesting to observe that present LES are unable to predict the secondary peak clearly visible in the present DNS or in previous experiments. This inability suggests that small-scale mechanisms, not captured by the LES mesh, are involved in this heat transfer feature at the wall. The investigation of the vortical structures explicitly computed by DNS seems to confirm this view, through a complex link between the distortion of big rolls highly coherent in the azimuthal direction (captured by LES) and the stretching of radial small-scale vortices produced between them (at a scale unreachable by LES mesh). This study is a first step toward the production of an extensive DNS database dedicated to turbulence modeling in the context of LES, RANS and hybrid RANS-LES methods. The code ‘‘Incompact3d’’’s ability to deal with a considerable number of computational cores (more than 200 000, see Laizet and Li, 2011) enables us to plan DNS and LES of the same flow configuration at higher Reynolds number. In this more realistic context regarding the industrial applications, an increase of the spatial resolution of LES (consistently with the increase of the Reynolds number) can hopefully lead to the correct capture of the secondary peak in the Nusselt distribution. If this favorable behavior is shown, it could be concluded that the present LES’s inability to correctly predict

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Please cite this article in press as: Dairay, T., et al. LES of a turbulent jet impinging on a heated wall using high-order numerical schemes. Int. J. Heat Fluid Flow (2014), http://dx.doi.org/10.1016/j.ijheatfluidflow.2014.08.001