Turbulent flow and convective heat transfer in a wavy wall channel

Transcription

1 Turbulent flow and convective heat transfer in a wavy wall channel A.Z. Dellil, A. Azzi, B.A. Jubran Heat and Mass Transfer 40 (2004) DOI /s Abstract This paper reports the numerical modeling of turbulent flow and convective heat transfer over a wavy wall using a two equations eddy viscosity turbulence model. The wall boundary conditions were applied by using a new zonal modeling strategy based on DNS data and combining the standard k e turbulence model in the outer core flow with a one equation model to resolve the near-wall region. It was found that the two-layer model is successful in capturing most of the important physical features of a turbulent flow over a wavy wall with reasonable amount of memory storage and computer time. The predicted results show the shortcomings of the standard law of the wall for predicting such type of flows and consequently suggest that direct integrations to the wall must be used instead. Moreover, Comparison of the predicted results of a wavy wall with that of a straight channel, indicates that the averaged Nusselt number increases until a critical value is reached where the amplitude wave is increased. However, this heat transfer enhancement is accompanied by an increase in the pressure drop. Nomenclature x i general non-orthogonal coordinate system J Jacobien of the coordinate transformation / time averaged variable C i convection term D / i diffusion term S / source term Kronecker delta d ij u 0 i u0 j u 0 j h k G ij Pr t S ij Reynolds stress tensor turbulent heat flux turbulent kinetic energy turbulent transport coefficient turbulent Prandtl number mean rate of strain Received: 21 October 2002 Published online: 17 October 2003 Ó Springer-Verlag 2003 A.Z. Dellil Faculté des sciences, Université dõoran Es-sénia, Algérie A. Azzi Faculté de Génie-Mécanique, Université des Sciences et de la Technologie dõoran, Algérie B.A. Jubran (&) Sultan Qaboos University, Department of Mechanical and Industrial Engineering, Muscat, Sultanate of Oman bassamj@squ.edu.om m t isotropic eddy viscosity C l model constant, =0.082 e rate of dissipation of k l l, l e length scales, eqs (7) and f(9) j von Karman constant, = 0.4 f l damping function, eq. (8) R y turbulent Reynolds number, eq. (10) q density H inter wall spacing am wave amplitude k wave length pffiffiffiffiffiffiffiffiffiffi U s friction velocity, ( s w =q) Re Reynolds number, (HU/m) 1 Introduction Wavy wall flows occur under a wide variety of engineering applications and have, consequently, received considerable attention [1 2]. One of the most important applications is the heat transfer enhancement in heat exchangers. The physical process of enhancing heat transfer in such application is to introduce some geometrical modifications on the wall in question in order to break the boundary layer that forms on the exchanger wall and replace it by a fresh fluid from the free stream flow [3]. In real applications, engineers are also interested in the additional pressure drop caused by such techniques. So, the best solution is that provides the least pressure drop and the largest heat transfer rate. Other parameters such as simplicity, manufacturability, maintenance, etc., are also important parameters in the design phase [3]. Wavy wall heat transfer enhancement technique has been used extensively in the design of compact heat exchangers as can be seen from the numerous experimental and numerical investigations reported in the literature. One of the important observations in this field is that in the laminar regime, where wavy passages provide significant heat transfer enhancement when the flow is unsteady. However, for steady flow, the enhancement is insignificant. Wang and Vanka [3] conducted a numerical investigation for laminar steady and unsteady cases with wavy wall. They reported a heat transfer enhancement factor of about 2.5 for the unsteady case compared with that for a parallel-plate channel case. However, for the steady case with wavy wall the heat transfer rate is only slightly improved. The transition occurs at a Reynolds number of around 180 for the geometrical configuration studied by Wang and Vanka [3]. 793

2 794 In most industrial process the flow is always turbulent. Consequently, wavy wall flows are receiving renewed attention. Recently, Hudson et al. [4] studied experimentally a rectangular water channel where the lower wall was constructed with removable Plexiglas plates on which the waves were milled. To insure that a fully periodic flow had developed, the LDV measurements were made above the 31 st of 36 waves. The wavelength and the amplitude were set to (k = H) and (am = 0.05 H), respectively, where H is the mean height of the channel. This experimental work is one of the few studies which provides extensive measurements of the Reynolds stresses. It was followed by numerical simulation of a fully developed turbulent flow over a wavy wall [5] and [6]. This type of turbulent flow is numerically very challenging, since it is characterized by periodic changes of the pressure gradient, curved streamlines, and for important wave amplitude, separation and reattachment can occur. The work reported by Maaß and Schumann [6] is largely documented, and hence it was selected as a benchmark to validate the dynamic part of the present computation. In addition to the dynamic field, the present paper reports a numerical parametric-investigation on forced convective heat transfer in a wavy wall channel. Detailed flow and thermal fields were obtained by solving RANS (Reynolds Averaged Navier Stokes equations) equations and a two-equation eddy viscosity turbulence model. The wall boundary conditions were applied by using a new zonal modeling strategy based on DNS data and combining the standard k e turbulence model in the outer core flow and a one equation model resolving the near-wall region. This strategy was initialized by Rodi et al. [7] and tested with success in previous computations [8]. 2 The mathematical model The mathematical model consists of the RANS, the twoequation eddy viscosity k e turbulence model and the energy equation. The governing equations for steady, turbulent, incompressible flows in non-orthogonal coordinates using Cartesian velocity components can be written in a generalized form as follows: C i / þ D / i ¼ S / ; ð1þ i Where / is the considered time-averaged variable, J is the Jacobian of the coordinate transformation, C i, D / i represent respectively the convection and the diffusion terms Table 1. Form of terms in the individual equations / C i D / i S / 1 b i jqv j 0 0 v k b i j qv j l J B k j þ b i j xj k j k b i jqv j l r k J B j P K qe e b i jqv j l r ek J B j B i j ¼ cofactor of i=@x j Þ; B i j ¼ bi l bj l x j k ¼ j l ; P k ¼ l t J b l n þ b j m b l n l ¼ l l þ l t ; l=r k ðl l þ l t =r k Þ; l=r e ðl l þ l t =r e Þ pb j k C e1 e k P k C e2 q e2 k and S / is the source term. Table 1 shows the above terms for each considered dependent variable; y i denotes a reference Cartesian coordinate system while x i is a general non-orthogonal coordinate system. The Reynolds-stress tensor and the turbulent heat flux are approximated within the context of the k e turbulence model and the near-wall viscosity-affected region is resolved with a one-equation model [7]. The two-layer approach represents an intermediate modeling strategy between wall function and pure low-reynolds number model. It consists of resolving the viscosity-affected regions close to the wall with a one-equation model, while the outer core flow is treated with the standard k e model. In the outer core flow, the usual eddy-viscosity hypothesis is used, applying a linear relation of the Reynold-stress tensor to the velocity gradient as follows: u 0 i u0 j ¼ 2 3 d ijk 2C ij S ij u 0 j h ¼ C ij@t ð3þ Pr j where d ij is the Kronecker delta, k u 0 i u0 j =2 is the turbulent kinetic energy, and S ij is the rate of strain tensor. For high- Re flows, the turbulent transport coefficient G ij is conventionally made isotropic and proportional to a velocity scale (3k) and a time scale (k/e), characterizing the local rate of turbulence and is given by: C ij m t C l k 2 =e where m t is known as the isotropic eddy viscosity, e represents the rate of dissipation of k, and C l stand for a model constant. The distributions of k and e are determined from the conventional model transport equations of Jones and Launder [9], and standard values can be assigned to the model constants. In this flow region the turbulent Prandtl number is usually fixed at 0.9. In the one equation model, the eddy viscosity is made proportional to a velocity scale determined by solving the k-equation, and a length scale l l prescribed algebraically. The dissipation rate e is related to the same velocity scale and a dissipation length scale l e, also prescribed algebraically [7]. Such model has the advantage of requiring considerably fewer grid points in the viscous sub-layer than any pure Low Reynolds scheme. Also, because of the fixed length-scale distribution near the wall, these models have been found to give better prediction for adverse pressure gradient boundary layer than pure k e models. The present two-layer model is a re-formulated version of the so-called ðv 2 Þvelocity-scale based model (TLV) proposed by Rodi et al. [7]. In a recent study Azzi and Lakehal [8] re-incorporate k 1/2 as a velocity scale instead ðv 2 Þ 1=2 and l l and l e are re-scaled on the basis of the same DNS data of Kim et al. [10]. This model will be call hereafter as TLV model and defined as follows: pffiffi C ij m t C l k ll ð5þ e ¼ k 3=2 =l e ð2þ ð4þ ð6þ

3 l l ¼ jycl 3=4 f l ð7þ f l ¼ 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0:116 R 32 y þ R y ð8þ l e ¼ jcl 3=4 y ð9þ 2 þ 17:28= f l R y jc 3=4 l p R y ¼ q ffiffi k y=l ð10þ Where j = 0.4 and C l = The outer and the near-wall model are matched at the location where f l = 0.95, indicating that viscous effects become negligible. More details can be found in [8], where the model is tested for a fully developed channel and applied for a film cooling configuration. 2.1 Numerical procedure The numerical procedure used to calculate the test case is based on a finite-volume approach for implicitly solving the incompressible Reynolds Averaged Navier Stokes equations (RANS) on arbitrary non-orthogonal grids, employing a cell-centred grid arrangement. The momentum-interpolation technique of Rhie and Chow [11] is used to prevent pressure-field oscillations and the pressurevelocity coupling is achieved using the SIMPLEC algorithm of Van Doormal and Raithby [12]. The resulting system of the algebraic difference equations is solved using the Strongly Implicit Procedure (SIP) of Stone [13]. The convection fluxes are approximated by employing the QUICK scheme for all variables applied in a scalar form by means of a deferred-correction procedure and bounded by the Van-Leer Harmonic function as limiter. The diffusive fluxes are, however, approximated using second-order central differences. Convergence was in all cases determined based on a drop in normalized mass and momentum residuals by four orders of magnitude. A global mass balance algorithm was employed after each iteration to correct the mass fluxes at the outflow. More details of the mathematical formulation, which are now a standard material and well known to most investigators, can be found in the following references [14 15]. 2.2 Computational domain and boundary conditions The computational domain consists of 10 waves preceded and followed by upstream and downstream flat sections for flow adjustment and recovery, respectively. This geometrical configuration was used previously by Patel et al. [2] and seems to be more exact than using one wave length of channel and applying periodic conditions. According to Patel et al. [2] the periodic boundary conditions cannot be established without prior knowledge of the flow, particularly with regard to the turbulent quantities and calculations of the type performed here. The wavy and the flat plate are placed with a mean spacing (H = 1). The amplitude and the wavelength of the lower sinusoidal wavy wall are (am = 0.05H) and (k = H), respectively. The upstream and downstream flat sections are 4k and 2k in length respectively. For the inlet conditions, a similar procedure to that used by Patel et al. [2] is applied here. The distributions of velocity and turbulence parameters in a fully developed flow at sufficiently large distance from the entry of 80H straight channel, which are independent of the initial conditions and invariant with distance, are used as inlet conditions for wavy-wall calculations. The Reynolds number was set as in DNS computations [6] at Re(HU/m) = The turbulence intensity is assigned a value of 5% and the turbulence dissipation is calculated based on a turbulent viscosity equal 50 times the laminar viscosity. At the outflow boundary, zero-gradient conditions are imposed for all dependent variables. 2.3 Grid mesh The quality of a computational solution is strongly linked to the quality of the grid mesh. So a highly orthogonalized, nonuniform, fine grid mesh was generated with grid nodes considerably refined in the near-wall region. The normalized y + values at the near wall node are less than unity, and care is taken so that the stretching factors are kept close to unity. Figure 1 shows a close-up of the computational grid used for the case with am = 0.10 H. The grid adopted for the computations was obtained after a series of tests and consists of 42,100 grid nodes (disposed on a global array nodes in x, and y directions). For the parametric study, several grids were generated, where the wave amplitude was varied from 0.0 to 0.10 by a step of Results and discussion 3.1 Comparison with DNS data In order to validate the code the first run was done with flow characteristics that are the same as those used by Maaß and Schumann [6]. Figures 2 and 3 compare the normalized mean velocity and turbulent kinetic energy profiles with DNS data. The comparison is done until the middle of the channel and at four streamwise locations of the seventh wave that correspond to the divergent, trough, convergent and crest positions, respectively (x/h = 0.304, 0.492, and 0.992). The figures show reasonable agreement for the mean velocity profiles. The reverse flow occurring in the first half of the wave (x/h = and 0.492) is clearly captured by the computations. However, the turbulent kinetic energy is globally underpredicted. As Fig. 1. Close-up of the computational grid for the case with 0.1 H of amplitude 795

4 796 Fig. 2. Comparison of normalized mean streamwise velocity with DNS data 6: x/h=0.304: divergent; x/h=0.492: trough; x/h=0.804: convergent; x/h=0.992: crest Fig. 3. Comparison of normalized turbulent kinetic energy with DNS data 6: x/h=0.304: divergent; x/h=0.492: trough; x/h=0.804: convergent; x/h=0.992: crest it was observed in an earlier test on flow over flat plate [8], this underprediction is a consequence of an overpredicted value of the turbulence dissipation in the region where k + reaches its peak value. However, the underprediction is more important in this wavy wall case than it was for the flat plate. It can also be observed that turbulent kinetic energy is represented by very different plots along the wave and until approximately 0.25 H normal to the wall. The maximum peak is observed at x/h = in the streamwise direction and roughly at 0.05H normal to the wall. Figure 4 shows the normalized logarithmic axial velocity and temperature profiles at trough and crest of the seventh wave. In the figure, U + and y + are computed with the friction velocity U s based on the local wall shear stress. The standard velocity and temperature logarithmic laws are also plotted in the figure. For comparison, the computed profiles for the flat plate and for the trough and crest of wavy wall (0.05 H) are presented. As it was found by Patel et al. [2], the most important observation is that a standard law is clearly not applicable in such complicated configuration. The strong adverse and favorable pressure gradients are responsible for the new trend shown by the plots. At the crest station, which corresponds to mildly favorable pressure gradient, the velocity and temperature distributions are underpredicted compared with the logarithmic law. At the second station, which corresponds to the trough, the strong adverse pressure gradient is Fig. 4. Law of wall for Velocity and temperature at trough and crest of the wavy wall, am=0.05 responsible for the overprediction of the velocity profile compared with that of the logarithmic law. Between the two stations, it can be seen and as expected that there is a zone where the profiles agree with the logarithmic law due to the near zero pressure gradient. 3.2 Influence of the wave amplitude The effect of geometric parameters is investigated by varying the amplitude wave from zero (flat plate) to 0.1 H by a step of 0.02 H. The friction and the pressure coefficients distributions for trough one wave in the fully developed region are plotted in figures 5 and 6, respectively. Comparing the results for flat plate (am = 0) with that obtained for the wavy wall, it can be seen that there is a decrease in the values of friction coefficient in the trough and an increase in the values in the crest. Except for very small amplitude (am = 0.02 H), the shape of the distribution curves is highly deformed with a tendency to increase values even in the trough. The separation and the reattachment points where the friction coefficient vanishes are plotted in figure 7. Separation and reattachment points correspond to locations where the friction coefficient vanishes. For (am = 0.5 H) they are at 0.13 k and 0.58 k, respectively. In DNS computations [6], they were located at 0.14 k and 0.59 k, respectively. The agreement is completely satisfying. It is also obvious from the figure that the

5 797 Fig. 5. Friction coefficient versus amplitude wave Fig. 8. Stream function and vector plot Fig. 6. Pressure coefficient versus amplitude wave Fig. 7. Separation and reattachment points versus amplitude waves reattachment point is more sensitive to amplitude waves than the separation point. The pressure coefficient shows also an undulated character with maximum in the vicinity of the trough and minimum nearly in the crest (opposite to that observed for the friction coefficient). It is obvious that the pressure drop is proportional to the amplitude wave (Cp is zero at the inlet of the channel for all cases). Figures 8(a) and 8(b) show vector plots and computed streamlines for wave amplitude of 0.05 H and 0.10 H, respectively. One can see that separation is observed for both cases. From figure 5, which shows the friction coefficient distribution, one can predict that separation occurs first, roughly at am = 0.03 H, since for am = 0.2 H the friction coefficient is always positive and reaches negative values for am = 0.4 H. We can also see that the recirculation zone increases in size and its centre moves in the downstream direction when the amplitude wave increases. The turbulence intensity contours for the corresponding geometry are plotted in figures 9(a) and 9(b). The maximum turbulence zone, which is located near the wavy wall, increases in intensity and moves in the downstream direction when the amplitude wave increases. This feature corresponds to what is expected from the wavy wall in perturbing the boundary layer and enhancing turbulent heat transfer. The temperature contours are presented for the same two amplitude waves in figures 10(a) and 10(b). Having in mind that zero value is assigned to the wall and unity value for the inflow, one can see that reattachment points are always recognized by high temperature gradient. We can also see that temperature contours, which are straight in the flat-plate channel, are distorted in the recirculation region when increasing the amplitude waves. This supports the fact that the boundary layer is perturbed by eddies. The variation of the local Nusselt number through one wave in the fully developed region versus the amplitude wave is presented on figure 11. The minimum and the maximum Nusselt numbers are located near the separation and the reattachment points, respectively. The difference between the maximum and the minimum values, increases with the amplitude wave. However, as shown in figure 12 the averaged Nusselt number increases with the

6 798 Fig. 11. Variation of local Nusselt number for various amplitude wave Fig. 9. Turbulence intensity contours (in %) Fig. 12. Variation of averaged Nusselt number versus amplitude wave Fig. 10. Contours of a dimensional temperature amplitude wave until 0.06H, after which the Nusselt number is approximately remains constant. 4 Conclusions Numerical results for the turbulent flow and heat transfer in a two-dimensional channel with wavy wall are presented. The wavy channel studied corresponds to the geometry of Maaß & Schumann [6]. Special attention has been focused on periodic, fully developed thermal and flow fields. Computations have been performed by a finite volume method, solving flow and energy equations. The Reynolds-stress tensor and the turbulent heat flux are approximated within the context of the k e turbulence model. The near-wall viscosity-affected region is resolved with a one-equation model. The predicted results are in good agreement with published DNS data. The two-layer model is found to be successful in capturing most of the important physical features of such a flow with reasonable amount of memory storage and computer time. The predicted results show the shortcomings of the standard law of the wall for predicting this type of flows and consequently suggest that direct integrations to the wall must be used instead. A geometrical parametric study was conducted by changing the amplitude-to-wavelength ratio. Comparison of predicted results of a wavy wall with that of a straight channel indicates that the averaged Nusselt number increases until a critical value is reached where the amplitude wave is increased. However, this heat transfer enhancement is accompanied by an increase in the pressure drop.

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