over cylinder at high Reynolds number
Cite as: AIP Advances 9, 095012 (2019); https://doi.org/10.1063/1.5118421
Submitted: 03 July 2019 . Accepted: 29 August 2019 . Published Online: 10 September 2019 A. Aravind Raghavan Sreenivasan , and B. Kannan Iyer
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Enhanced wall turbulence model for flow over cylinder at high Reynolds number
Cite as: AIP Advances9, 095012 (2019);doi: 10.1063/1.5118421 Submitted: 3 July 2019•Accepted: 29 August 2019•
Published Online: 10 September 2019
A. Aravind Raghavan Sreenivasan and B. Kannan Iyera) AFFILIATIONS
Mechanical Engineering Department, IIT Bombay, Mumbai 400076, India
a)Author to whom correspondence should be addressed. Electronic mail:[email protected]
ABSTRACT
Modeling of turbulent flow over cylinders at high Reynolds numbers continues to be a challenge despite extensive work available in the literature. Most models suffer from loss of accuracy or require extremely refined grids both of which render their usage very difficult for practical problems. A wall model has been developed for solving supercritical turbulent flows using the Turbulent Boundary Layer Equations (TBLE). A new way to calculate the shear stress using the model has been introduced in this work. The TBLE model requires an input velocity from the off wall Large Eddy Simulation (LES) model. The method to obtain the same has been devised using the Log-Law. Also, the calculation of the turbulent viscosity in the near wall region has been modified by varying the Von Karman coefficient as a function of velocity in the adverse pressure gradient region. The results obtained with this enhanced TBLE model have been compared with other popular turbulence models for a Re of 1.0×106. The TBLE model has then been used to solve two more Re of 6.5×105and 2.0×106. The performance of the model has been compared with respect to mean drag coefficient, Root Mean Square (RMS) of lift coefficient, Strouhal number, base pressure coefficient, adverse pressure recovery and separation angle as well as the profiles for pressure and shear stress variation over the cylinder. The model is shown to be fairly accurate, robust and computationally efficient on account of its ability to work with relatively coarse grids.
© 2019 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).https://doi.org/10.1063/1.5118421., s
I. INTRODUCTION
Flow over bluff bodies is one of the most commonly occur- ring flows in nature as well as in several industrial applications. Such flows are encountered in aviation, automobile, refrigeration, power plants and civil engineering. The flow over a cylinder has often been used to study flow physics because of the canonical nature of the problem. Even though the geometry is quite simple, modeling it is still challenging as there is attached flow, separated flow, recircu- lation, vortex shedding, etc. The flow patterns also keep changing as the Reynolds number (Re) of the upstream flow increases. As mentioned in Zdravkovich,1the flow physics change from creeping flow at very low Re (Re<5) to trans-critical flows at very high Re (Re > 6 × 106). Beyond Re of 200, the flow starts to become 3 dimensional and turbulence sets in the far-wake region. For Re beyond 400, turbulence now sets in the shear layer. As the Re increases, the onset of turbulence keeps shifting upstream. At a certain Re, turbulence sets in within the attached boundary layer
leading to delayed separation resulting in a sudden drop in drag coefficient known as the “Drag Crisis”. Again there are several sub- regimes with the pre-critical flow in the range ofRe= 2×105to supercritical flow atRe= 1× 106. Beyond the supercritical range (Re>6×106), there is a recovery of drag as the separation point gets advanced. While there may be slight variations in the flow regime classification and the corresponding transition Re in the lit- erature, it is an accepted point that as the flow physics constantly changes modeling of the flow is quite challenging. A single satisfac- tory model for all ranges of Re is still elusive despite many decades of research.
Flow over cylinder often leads to vibrations which result in various kinds of problems ranging from noise to mechanical fail- ures. Many of these problems are fluid-structure coupled problems, where computational efforts need to be used judiciously. At the same time, it is important to obtain a fairly accurate estimate of the fluid forces as they, in turn, control the dynamics of the structure. The computational requirements for solving the flow domain increase
with increasing Re and most commercial problems occur at high Re.
Thus, there is a strong need for a turbulence model which can resolve high Re flows at reasonable computational costs.
The Reynolds Averaged Navier Stokes (RANS) models which are popularly used in industries are not that efficient in predicting massively separated flows as they tend to overpredict the turbu- lent stresses.2,3However, they are still commercially used consider- ing that they can solve the attached flow region without requiring grids as fine as those required for Large Eddy Simulations (LES) or Direct Numerical Simulation (DNS) based solvers. Despite the increasing availability of computational resources, it is still not prac- tically possible to do a DNS for high Re flows. Also, for most indus- trial problems, the timelines involved in obtaining a solution can- not be met by these kinds of simulations. LES, which evolved as an alternative to DNS, tries to resolve the larger, energy contain- ing, scales fully while modeling the smaller scales known as the sub-grid scales. The modeling of the sub-grid scale stresses is the main point of comparison between the different variants of the LES model.
While the traditional LES model has been quite popular in resolving the separated flow regions, for near-wall regions the grid spacing requirements scale as Re1.76.4 The traditional Smagorinsky model was used by Lilly to derive a value for the Smagorinsky coeffi- cient assuming isotropic turbulence. It was observed that the model sometimes led to the prediction of excessive damping near the wall.
To overcome this, Germano et al.5 introduced a dynamic model which used a double filtration approach to determine the values of the Smagorinsky coefficient dynamically as a function of space and time. The Wall-Adapting Local Eddy viscosity (WALE) model by Nicoud and Ducros6uses the strain rates to get the variation of the Smagorinsky coefficient with the main advantage of better predic- tion of near-wall behavior. Of late, the focus has shifted towards developing hybrid models which utilize the better resolution of LES away from the wall and better modeling capabilities of RANS near the wall. The Detached Eddy Simulation (DES) model, first devel- oped by Spalart,7 and applied by Travin et al.8 for a flow over a cylinder, uses some form of blending function which controls the RANS and LES regions of the flow. One of the main challenges with the hybrid models is their sensitivity in determining the region where the switching happens from LES to RANS. In certain models, this is specified by the user while in certain others there are blending functions which enable this switching based on the grid. However, these create challenges like “log-layer mismatch” and difficulty in achieving grid independence.9
As a different approach, there have been techniques devised to augment the existing LES model with a special wall model to deal with the increased near-wall computation requirements. A good review of the existing models has been carried out by Larrson et al.9These models are based on solving a filtered momentum con- servation equation in the wall parallel direction also known as the Turbulent Boundary Layer Equation (TBLE) which is given as follows:
∂u
∂t +∂uuj
∂xj
+1 ρ
∂p
∂x= ∂
∂y[(υ+υt)]∂u
∂y (1)
In Eq. (1), x refers to the wall parallel direction and y refers to wall normal direction. Additionally, most wall models also assume a mixing length model for the turbulent viscosity
given by:
υt
υ =κy+(1−exp(−y+/A))n (2) Further, an equilibrium assumption is considered between the unsteady, convective and pressure gradient terms and thus the left hand side of Eq.(1)becomes zero. While this is true for steady flows at the edge of the boundary layer, it acts as a simplifying assumption for non-equilibrium flows like the flow over a cylinder. However, as it helps to bring down the computational costs significantly, it has been employed for non-equilibrium flows as well. As will be shown later, this assumption holds true for most of the attached flow region and the predictions using this approach are quite satisfactory. Thus, Eq.(1)is reduced to an Ordinary Differential Equation (ODE) which is then solved numerically to obtain the wall stress. Alternatively, the ODE is integrated along the wall normal direction to yield an alge- braic non-linear equation which is then iteratively solved to obtain the wall stress. The major advantage of solving it algebraically is the reduced computation time while ODE solvers can handle any additional physical effects like compressibility etc. in case they are present in the equation.
The earliest attempt to use the TBLE was done by Balaras et al.10who solved the complete Eq.(1)as a Partial Differential Equa- tion (PDE) without the equilibrium assumptions. Catalano et al.11 used a slightly modified form of Eq.(1)where they dropped the unsteady and convective terms but retained the pressure gradient.
The model was used to solve for flow over a cylinder at a supercritical Re of 106. While it was able to predict the drag coefficient and pres- sure distribution reasonably, the skin friction prediction was very much overestimated as compared to experimental results. Again, the model required the velocity to be input from the first off wall node which can lead to loss of accuracy as the LES model may not be able to predict the velocity correctly in that region as pointed out by Larrson et al.9There have also been several other attempts in the literature using the wall stress models but most have been restricted to channel flows and lower Re.
From the discussions thus far, it is clear that there is a need to have a wall model which can give reasonably good predictions at high Re. While some amount of grid sensitivity cannot be avoided due to the physics of the boundary layers, the model should not give drastically different results with slight modifications of the bound- ary layer grid. Further, the user inputs to the model must be kept minimum i.e. the model should pick the appropriate region from where the velocity can be picked as an input to calculate the wall stress. Keeping the above in mind, an attempt has been made to modify the wall stress model to enable an accurate prediction of flow at supercritical Re. The velocity input for the TBLE model has been estimated using an iterative methodology considering a log profile assumption. Furthermore, the value of the Von Karman coefficient in the separated flow region has been calculated dynamically using the velocity profile. While the log-law is often used in wall functions, the use of this in conjunction with the TBLE model as well as a mod- ified treatment of separated flow regions in the wall stress calculation is a novel attempt.
At these Re, the most commonly available results are engineer- ing parameters like drag coefficient and Strouhal number. There is, however, a significant scatter in the experimental measurements.
The wide variations obtained for the drag coefficient reported in the literature has been attributed to variations in the influencing
FIG. 1. Variation of CD with Re.
(solid squares) Achenbach,12 (hollow triangles) Bursnall and Loftin,13 (hollow circles) Delany and Sorenson,14(hollow diamond) Schewe,15(dashes) Spitzer,16 (pluses) Weiselsberger,17 (hollow squares) Shih et al.,18 (---) Re solved in current work.
parameters like free stream turbulence, surface roughness, blockage ratio, etc.1 Even slight variations in these can result in significant deviation in the results as the disturbances get amplified due to the high level of turbulence involved at the current Re. The experimen- tally obtained distribution of the drag coefficient as a function of Re by several investigators (Achenbach,12Bursnall and Loftin,13Delany and Sorenson,14Schewe,15Spitzer,16Weiselsberger,17Shih et al.18) is shown inFig. 1.
As seen fromFig. 1, at certain Re the values of the drag coeffi- cient measured by different experiments differ by more than 100%.
The high scatter makes it difficult to identify appropriate experi- ments for benchmarking the numerical solver. As mentioned earlier, one needs to ensure that the experimental results being used to com- pare the numerical case have similar influencing parameters. This was taken into consideration while selecting experiments for com- parison which will be explained further in the Results and Discussion section.
The paper is organized as follows. The details of the wall model used have been explained in the next section followed by the details of the numerical setup. The results using the TBLE models have been obtained at Re of 6.5× 105, 1.0 ×106 and 2.0×106 which lie in the turbulent boundary layer flow regime. These are elabo- rated in the Results and Discussion section. Apart from drag coef- ficient, other parameters of interest like the base pressure coef- ficient, separation angle, adverse pressure recovery, RMS of Lift coefficient and Strouhal number have also been compared with the appropriate experimental results wherever available. The dis- tribution of pressure and shear stress over the cylinder, have been studied through the pressure coefficient and the skin-friction coef- ficient variation respectively. For Re = 1.0 ×106, the model has been compared with popular models of LES like the Smagorinsky- Lily (SL) model and the WALE model apart from a RANS (k-ε) model, and hybrid models (DES (SA) and SAS models). The sum- mary sums up the efforts and also presents a comparison of the
computational times. Finally, the conclusion section presents the key findings.
II. ENHANCED WALL TURBULENCE MODEL A. Methodology
As mentioned in the previous section, the wall stress model is based on the solution of the momentum conservation equation given in Eq.(1). By taking the derivative of the RHS using the product rule we get,
∂u
∂t +∂uuj
∂xj +1 ρ
∂p
∂x =∂u
∂y× ∂
∂y(ν+νt)+(ν+νt) ×∂2u
∂y2 (3) If one considers equilibrium in the boundary layer flow, then the convection and pressure gradient terms can be assumed to cancel each other. To verify this assumption, the sum of the LHS terms of eq.(3)was divided by the first term of the RHS of eq.(3)for the case of Re = 1.0×106in the near wall region. This ratio (α) is given by
α= ∣(∂u
∂t +∂uuj
∂xj
+1 ρ
∂p
∂x)/[∂u
∂y× ∂
∂y(ν+νt)]∣ (4) The variation ofαover the cylinder is shown inFig. 2. As can be seen, the value ofαis negligible for most of the attached flow region.
Despite the peaks seen in the separated flow region as well as near the front stagnation point, the shear stress in these regions are fairly low and thus, the equilibrium assumption will not have any significant impact on the shear stress calculations. Thus the left-hand side of Eq.(1)becomes zero and it reduces to
d
dy[(ν+νt)]dux
∂y =0 (5)
FIG. 2. Typical variation of ‘α’ over the cylinder in the enhanced TBLE model for Re = 1.0×106.
Where,
ux= Wall parallel velocity
Integrating the above equation along the wall normal direction from the wall up to the edge of the boundary and rearranging them we can obtain an expression for the wall shear stress as follows,
τwi=ρuLES/ ∫0δ 1
(ν+νt)dy (6) The velocity,uLES is obtained from the LES model. The value of turbulent viscosity is obtained using a zero-equation damped eddy viscosity model19,20 as given in eq.(2). ‘A’ and ‘n’ are constants which are taken differently by different researchers based on their wall model. Wang and Moin19,20took A=19 and n=2 for their TBLE model. In this work, the value of ‘A’ was taken as 25 and ‘n’ as 3 in order to ensure the eddy viscosity reduced to zero as one approached the laminar boundary layer (y+<5). ‘κ’ is a model coefficient whose value is either considered as 0.420,11or obtained dynamically.19The effect of the model coefficient is explored further subsequently. Due to the dependence of eddy viscosity on y+which itself is a function of wall shear, the equations need to be iteratively solved to obtain the correct wall shear.
Thus, in order to obtain the correct wall shear, it becomes imperative to use the correct velocity and correct prediction ofνt. In Wang and Moin’s model,20the velocity in the first off-wall node was considered as the input velocity. While it is a natural assumption, it brings with itself certain inaccuracies.9If the wall grids are not fine enough then they do not resolve the velocity correctly and using such a velocity will yield inaccuracies in the shear stress as well. Also, there is no guarantee that the assumed velocity satisfies the log law near the wall. Hence a better approach would be to consider the velocity at a certain distance from the wall which is clearly in the turbulent boundary layer where the log-law can be considered as fully applica- ble. Since the log-law is generally suited for wally+≥40, those cells which satisfy the above condition can be used to obtain the velocity.
As it cannot be knowna priorias to where these would lie, for every iteration, a radial sweep is required from the cell closest to the wall
till we reach the cell satisfying the log-law criterion. The information of the radial neighbors is obtained and stored before beginning the solution. The only condition is the requirement of structured grid layers near the wall. However, for most cases, even if the overall mesh is unstructured, there are always a few layers of structured boundary mesh near the wall to capture the boundary effects and hence it is not a major restriction. Once the cell is identified, the velocity of its face is interpolated using the log law equation of the form:
ux/uτ=A×logy++B (7) The constants are obtained using the cell velocities of the selected cell and it’s radially outward neighbor. This ensures both the cell velocities used for interpolation are satisfying the log-law cri- terion. Thus the interpolated velocity also satisfies the log-law. The next influencing factor is the model coefficient ‘κ’. For the attached flow, the universal velocity profile is taken as
ux/uτ= (1/κ) ×logy++B (8) Where,κ= von Karman coefficient. (Typically taken as 0.41)
While most researchers consider the Von Karman coefficient to be constant, it is interesting to note that it is based on the linearity assumption of the Prandtl’s mixing length theory where the turbu- lent length scale was considered to be directly proportional to the wall distance.
lm∝y (9)
Where,
lm= eddy length scale
y= Normal distance from the wall Thus,
lm=κ×y (10)
However, it was found that the eddy viscosity computed based on the above length scale was over-predicted in the near wall region.
Thus, the Van Driest Damping function, as shown in Eq.(2), was proposed to ensure the eddy viscosity exponentially decayed in the near wall region.
While the constant value of ‘κ’ is acceptable in the attached flow region, it cannot be extended with certainty near the point of separa- tion as well as in the post separation region. Hence, an attempt was made in this model to determine the value ofκdynamically in these regions. Considering the form of the Log-Law shown in Eq.(8), the value ofκwas evaluated from the velocities in the turbulent region.
The value of Von Karman coefficient was kept constant as long as the pressure gradient along the flow direction was favourable. As soon as the pressure gradient became adverse, the value ofκwas dynamically obtained. Thus,κwas evaluated as
κ=0.41,(favorable pressure gradient)
=Min(∣uτ×log(y2/y1)/(u2−u1)∣, 0.41), (Adverse pressure gradient)
(11)
Where, y2, y1 and u2, u1 are the wall normal distances and the wall tangential velocities of the two selected cells respectively. Even though the pressure gradient turns adverse, actual separation occurs a little more downstream. In this zone, the value ofu2−u1becomes quite small and the value ofκshows a sudden jump. A minimum criterion was imposed in order to ensure there are no unphysi- cal values of κwhich might occur in this zone. This value of κ is used in the estimation of the eddy viscosity as per the expres- sion given in Eq.(2). This is the condition used in the enhanced model which will be hence-forth be referred to as the ‘TBLE’
model.
A typical variation of κwith the formulation in Eq.(11) is shown in Fig. 3. It is interesting to observe that the value ofκ decreases substantially till the point of separation. The value ofκ then increases once the flow re-attaches and then fluctuates till the rear stagnation point. Asκdepends on the friction velocity, its pro- file closely resembles the variation of the wall shear stress once the pressure gradient turns adverse. While the TBLE as given in Eq.(5) is based on an equilibrium assumption, the effect of the adverse pres- sure gradient and the decelerating flow is incorporated indirectly through the evaluation ofκ. The comparison of varying the Von Kar- man coefficient vis-à-vis keeping it constant has been carried out in the results section. The model where the value ofκis kept constant is indicated as the ‘TBLE_k’ model.
FIG. 3. Typical variation of ‘κ’ over the cylinder in the enhanced TBLE model for Re = 1.0×106.
Thus, the model ensures that the wall shear stress is computed using the correct inputs and thus one can hope to achieve greater accuracy. Also, since the point of application of the model is dynam- ically determined, there is no restriction imposed on the first off wall grid. Hence there is no need for extremely fine grids near the wall which will result in saving computational time considerably, especially, for high Re flows. The calculation of the eddy viscosity outside the boundary layer was done as per the SL model formula- tion. All the simulations were carried out by using ANSYS-FLUENT (Release 14.5) and the TBLE model was executed through a User Defined Function (UDF). The details of the grid and case setup are mentioned in the subsequent section.
B. Problem setup
The domain used for simulation is usually scaled by the length scale of the problem which happens to be the cylinder diame- ter (D) in this case. The domain and the boundary conditions should be such that they do not interfere with the flow around the cylinder. Different researchers have used different dimensions. The inlet turbulence was kept as zero for all the models except the k- ε model where 0.1% turbulent intensity was taken without which vortex shedding did not get generated. As the flow conditions were expected to remain fairly uniform till the cylinder is encountered, the domain length from the entrance up to the cylinder is shorter than other domain lengths. The exit boundary should be far from the source of the disturbance so that the gradients in the flow direc- tion become minimum at the exit and so there is little impact of the exit boundary. The transverse boundaries tend to have a significant impact on the flow and so it is often advised to keep them fairly large. While the flow is 3 dimensional, the gradients in the span-wise direction are much lesser as compared to the in-flow and trans- verse directions. The span-wise boundaries are usually of the order of 3D-4D21as an appropriate compromise between reducing com- putational time while maintaining reasonable accuracy. Catalano et al.11had argued that reduced correlation lengths in the span-wise direction at higher Re meant that even a shorter span-wise domain length would be sufficient. The span-wise length was taken to beπD in the current simulations. The final domain used is shown inFig. 4.
The boundary conditions used are as follows:
Inlet: Velocity Inlet (normal velocity is specified)
Outlet: Outflow Conditions (Gradient in the flow direction is taken as zero)
Top and Bottom of Domain: Symmetry boundary condition (free slip, no penetration condition)
Cylinder: Wall condition (no slip, no penetration condition) Spanwise: Periodic Boundary Condition
Two sets of grids were considered, an initial coarse grid which was later refined near the cylinder and in the span-wise direction.
The coarse grid (Grid 1) had 1.6×106elements, while the fine grid (Grid 2) had 3.3×106elements. The distribution of grids in the wall normal direction (y) was done using O-grid with exponential bunch- ing methodology chosen in such a way that there is a greater mesh density near the wall as shown inFig. 5. The expansion ratio in the near wall region was around 1.06 for both the grids. The idea is to create a grid such that there are at least 400 points capturing the cir- cular profile of the cylinder accurately to account for the gradients in
FIG. 4. Computational Domain used for simulating flow over a cylinder.
the flow direction due to flow acceleration, retardation, separation etc. The wall normal grid is then chosen such that the near wall grids have an aspect ratio of around 15 for the finest wall normal grid.
There were around 40 points considered in the span-wise direction for the coarse grid (Grid 1) which was increased to 60 for the fine grid (Grid 2). Catalano et al.11had used around 50 points for their grid while Travin et al.8took 30 grid points for the cases involving turbulent separation.
The typical variation of the near wall grid spacing in terms of non-dimensional wall units for the two grids is shown inFig. 6 andFig. 7 respectively. The shear stress obtained using the TBLE model for the corresponding grid has been used to compute the non- dimensional wall units. Considering that the enhanced turbulence model does not require fine grids near the wall, the grids were kept quite coarse in the wall normal direction for Grid 1. The Grids were
FIG. 5. Near-wall refinement for Grid using exponential distribution (Grid 2).
then refined in order to test the effect of refinement on the model performance. Even the fine grid used here is coarser than the ones, traditionally used for LES model without wall functions. These were considered to judge the efficacy of the wall model in making signifi- cant improvements in the performance of the LES models. The wall units considered here are similar to the ones used by Catalano et al.11 for their TBLE model at the same Re.
An additional requirement for the current model was the avail- ability of grid information in the radial direction. Hence while solv- ing the problem through parallel computing, it is necessary to par- tition the mesh surrounding the cylinder in a radial manner. This ensures that every computational node carries the information of the radial neighbors of the wall elements. The contour showing the near wall mesh colored as per the cells in each computational node is shown inFig. 8for 12 nodes
A ‘convergence-criteria’ of 1× 10−4 was maintained for the scaled residues obtained from the solution of the Continuity and Momentum equations as defined in ANSYS FLUENT (Release 14.5).22This was maintained at every time step to determine conver- gence and the flow converged at every time step. While the number of iterations for convergence varied with flow time, once the flow settled, typically the flow converged with 3-4 iterations per time step.
A non-dimensional time step given byΔt∗ = (ΔtU∞)/D was used to decide the desired time-step. For the current simulation, in order to optimize the computation time,Δt∗was taken to be 0.003.
It was found that there was no change in the results even if the time step was decreased by a factor of 2. The steady state potential flow solution was used to initialize the flow. This ensured that the initial transients were stabilized faster and the flow settled into its relatively periodic nature. The sampling time is given by
T∗= (Tf×U∞)/D (12) Where,
Tf= Flow time at the end of sampling (sec)
Sampling was initiated at around T∗=30. By which time, the initial transients had reasonably subsided. However, the k-εmodel required a higher time to settle and so the sampling was initiated at T∗= 50. However, the model showed a fairly regular vortex shed- ding after that without much variation in the mean values. Typically, the sampling time (T∗) is taken in such a way that the time statistics are relatively invariant. For Grid 1, results for different models at Re = 1.0×106were compared for different sampling times of 100, 200 and 300. It was found that while there were slight variations for some models, they were insignificant as compared to the overall dif- ference between the results predicted by the different models. Hence, the subsequent results have been obtained with a T∗=100. The momentum and the transport equations were solved using a second- order method while a Bounded implicit second-order method22was used in solving the transient problem. The SIMPLE algorithm was used as the solution algorithm. The results obtained for these models is given in the subsequent section.
III. RESULTS AND DISCUSSION
The simulations were first carried out at a super-critical Re of 1×106where the performance of the TBLE model was compared with those of other existing models. The TBLE models were then
FIG. 6. Variation of non-dimensional wall distance in the wall tangential (x), wall normal (y) and span-wise directions (z) for Grid 1.
FIG. 7. Variation of non-dimensional wall distance in the wall tangential (x), wall normal (y) and span-wise directions (z) for Grid 2.
used to solve flows at Re = 6.5×105as well as Re = 2×106. Param- eters like mean drag coefficient, Strouhal number, base pressure coefficient, separation angle, etc. and profiles of pressure coefficient and skin friction distribution have been compared with relevant experimental data available in the literature. The performance of the models with respect to different grids has also been evaluated.
The current simulations have been performed for a smooth cylin- der without any inlet turbulence. The flow conditions are symmet- ric and flow is considered reasonably ‘disturbance-free’.1Consider- ing the above factors in mind, the experimental results chosen for comparison have been ones with low free stream turbulence, larger L/D ratio, low blockage ratio, smooth cylinders (or very low rough- ness). The results of Schewe,15 Shih et al.,18 James et al.,23 have been used for comparison of most parameters. The pressure pro- files of Warschauer and Leene24and Flachsbart25have been addi- tionally considered for pressure coefficient profiles at relevant Re.
Very few measurements of skin friction profiles at high Re are avail- able and the most commonly used results are those obtained by Achenbach.12However, Achenbach used a very short cylinder (L/D
= 3.3) and the drag coefficient measured were higher than most other
FIG. 8. Contour of near cylinder grid (Grid 1) colored by the cell partition on 12 computational nodes.
TABLE I. Summary of influencing parameters for experiments considered for validation.
Re-range % freestream Blockage Surface
Authors (×106) turbulence Ratio (%D/H) L/D Roughness
Schewe15 0.023-7.1 0.4 10 10 Smooth
Shih et al.18 0.3-8.0 0.08 11 10.75 Smooth
James et al.23 0.14-10.9 0.2 9 24 Low Surface roughness
(K/D = 1.85×10−6)
Warschauer & Leene24 0.4-4.0 0.1 5.8 17.2 Smooth
Achenbach12 0.6-5.0 0.7 16 3.3 Low Surface roughness
(K/D = 1.33×10−5)
experiments as seen inFig. 1. However, they have still been plot- ted for comparison in the absence of any other data. A summary of the various influencing parameters for the experiments is shown in Table I.
A. Flow over the cylinder at Re = 1.0×106
As mentioned earlier, one of the objectives of the current work is to compare the performance of the TBLE models (varying as well as the constantκ) vis-à-vis some of the existing turbulence mod- els. As the current wall model uses the Smagorinsky-Lily (SL) model to solve for the flows outside the boundary layer, results were also obtained by using the LES-SL model without the wall model. The WALE model which was derived as an improvement over the SL model for predicting the near-wall behavior has also been used to solve the flow. Among the hybrid models, the SAS model, as well as the DES model with the Spalart Almaras (SA) as the RANS mode solver, have been used for comparison. Further, the k-εmodel with non-equilibrium wall functions26 has also been used to obtain a solution for the same problem.
The drag coefficient (CD) is of great engineering importance as it gives an estimate of the force the structure will be subjected to in the streamwise direction. As mentioned earlier, it is a function of Re and for the Re under consideration, the drag coefficient starts recovering slightly after reaching a minimum value during the ‘drag- crisis’. The variation of the drag coefficient for grid 1 as a function of sampling time is shown inFig. 9.
For this flow regime, the vortex shedding is irregular1and so one does not get a regular time signal for the lift and drag forces.
The TBLE models are initially very close to each other but gradually separate with the constantκmodel predicting a higher drag. The irregular nature of time signal is seen in the time signatures of all the results except the k-εmodel. The amplitude of the variation is also very small as compared to the other models. This can be attributed to the dissipative nature of the RANS based models which has been observed by Catalano et al.11as well. Despite the irregular nature of the time signals, the mean values do not change significantly over time for most of the models except the WALE model. For the grid under consideration, the WALE model shows a significant devia- tion in the results as compared to all the other models which can be attributed to the sensitive nature of the WALE model with respect to near-wall resolution. It is also interesting to note that apart from the TBLE models, all the other models predict a higher value for the
drag coefficients with the k-εmodel predicting a value in between them.
While the drag force keeps varying with time, the time- averaged values are considered here to indicate a mean force. The variation of the mean drag coefficient with respect to sampling time is shown inTable II. As seen fromTable II, most of the models do not show any variation for the mean drag coefficient with increasing sampling time. The WALE and the TBLE models show a variation of around 8% for the values at T∗=100 to T∗=300. However, the TBLE does not show any variation from T∗=200 to T∗=300. Consid- ering the significant variation in the values predicted by the different models, in order to assess the model performance, it is sufficient to consider the values at T∗=100 itself for comparison with exper- imental results without significant loss of accuracy while still saving substantial computation time.
The comparison of the Drag values for the two grids vis-à-vis the experimental results is shown inTable III. The numerical results of the LES with TBLE model obtained by Catalano et al.,11is also given for reference. It must be noted that the transition from the symmetric bubble regime to the supercritical regime happens over a range of Re1 and so the experimental results are also given for different values in the vicinity of the Re solved in the numerical
FIG. 9. Variation of drag coefficient (Grid 1) for different turbulence models with sampling time.
TABLE II. Variation of mean drag coefficient (CD) with Sampling Time for Re = 1
×106(Grid 1).
Turbulence CD
Model T∗= 100 T∗= 200 T∗= 300
SL 0.53 0.53 0.53
WALE 1.29 1.23 1.19
TBLE 0.25 0.23 0.23
TBLE_k 0.25 0.25 0.25
SAS 0.56 0.56 0.56
DES_SA 0.66 0.65 0.65
k-ε 0.40 0.40 0.40
work. Apart from the TBLE models, none of the other models pre- dict the drag coefficient within the experimental measurements for Grid 1 which indicates that the TBLE model is able to predict the fluid force fairly well even with a coarse grid. Once again, given the nature of the modeling, it is very difficult to have a model which is entirely unaffected by grid resolution.8,9However, among the mod- els considered here, the TBLE model shows minimum variation in the results due to grid resolution which indicates the robustness of the model. The TBLE with constantκshows more variation than the varyingκmodel. As the LES_SL model itself shows an increase in the drag coefficient, the TBLE models which depend on the LES model outside the boundary layer also show a similar trend. The WALE model showed maximum improvement due to grid refine- ment although the drag values predicted by it are still higher than the experimental results. The hybrid models are not able to predict the drag coefficient accurately even with a finer grid. The DES_SA TABLE III. Comparison of the mean drag coefficient (CD) with experimental results for Re = 1×106.
CD
Turbulence Model Grid 1 Grid 2
SL 0.53 0.58
WALE 1.29 0.32
TBLE 0.25 0.27
TBLE_k 0.25 0.30
SAS 0.56 0.74
DES_SA 0.66 0.48
k-ε 0.40 0.36
LES11 0.31
Shih et al.18
(Re = 8.5×105) 0.21
(Re = 1.2×106) 0.28
Schewe15
(Re = 8.9×105) 0.25
(Re = 1.3×106) 0.26
James et al23
(Re = 9.2×105) 0.16
(Re = 1.8×106) 0.30
model does give a better result than the SL model for the finer grid but it is still not close to the experimental predictions. The drag pre- diction by the k-εmodel is better than the other LES and hybrid models for both the grids which are due to the better near-wall mod- eling capabilities of the RANS model for coarse grids due to the use of the non-equilibrium wall functions.
The time history for the lift coefficient (Cl) is shown inFig. 10.
The lift coefficient shows a bit more regularity as compared to the drag coefficient but there is still scatter seen as the energy is dis- tributed over multiple frequencies for this flow regime. The TBLE models as well as the SL model show much smaller amplitude for theClvariation while the WALE model shows a much larger ampli- tude as it did for the drag coefficient as well. The hybrid models show similar behavior while the k-ε model shows a fairly regular variation of the lift coefficient once again. The mean lift coefficient for all symmetric flows is very small and hence the RMS value is often considered for comparison. The variation of the RMS of the lift coefficient (Ĉl) with sampling time is shown inTable IV. Once again, as compared to differences in the predictions for the different models, the variation in the value ofĈl for increasing sampling- time is quite small. Hence, we use the values obtained at T∗=100 for comparison with available experiments for both the grids as shown inTable V.
Among the experiments considered in this work for compar- ison, the RMS values of the lift coefficient were measured only by Schewe.15While all the models seem to over-predict as compared to the experimental values, the measurements indicate that the low val- ues could be seen in the symmetric bubble regime whereas a slightly higher value occurs once the flow transitions to super-critical flow.
Once again, the TBLE models seem to be closest as compared to all the other models for both types of grids. As compared to the mean drag coefficient,Ĉl shows much higher sensitivity to grid refine- ment. The Strouhal number (St) indicates the frequency of fluctu- ation of these forces which becomes an important parameter in the study of flow-induced vibrations. An FFT analysis of the lift force signal was used to determine the dominant frequency. The varia- tion of St with sampling time is given inTable VI. There is little influence of the sampling time on the Strouhal number. The com- parison of the St with experimental results for both the grids is given inTable VII.
For most of the models, the Strouhal number shows an increase with grid refinement which is particularly large for the WALE model. For super-critical flows, the St is expected to increase from an almost steady value of 0.21 which extends over a large range of Re. The TBLE model with varyingκdoes not show any change in the values with grid refinement while the constantκmodel shows a slight decrease. There is a lot of scatter in the experimental measure- ments (0.26-0.46) of St due to the irregularity of vortex shedding seen in this regime. Most models predict a value close to this range for the fine grid while k-εand the TBLE model predictions are within range for both grids.
While the drag and lift coefficients along with the Strouhal number are most commonly measured in experiments, there are other parameters which give us more insight into the nature of the flow around the cylinder. The base pressure coefficient (Cpb) is the value of the pressure coefficient at the rear stagnation point which has a strong influence on the overall drag coefficient. The adverse pressure recovery (Cpb−Cp,min) gives an indication of the
FIG. 10. Variation of lift coefficient (Grid 1) for different turbulence models with sampling time.
nature of the boundary layer along with an estimate of the angle of separation (θSep). Shih et al.18 calculated the angle of separa- tion by taking it as the average of the angle whereCp,minoccurred and the angle where the pressure coefficient starts to flatten out.
In order to compare with the measurements, θSep has been cal- culated using the same methodology for the numerical solutions.
For turbulent boundaries, the boundary layer can withstand higher adverse pressure gradient and thus the pressure recovery is higher and the separation angle is delayed. The variation of these parame- ters with sampling time for Grid 1 is given inTables VIII,IXandX respectively.
TABLE IV. Variation of RMS of lift coefficient (Ĉl) with Sampling Time for Re = 1
×106(Grid 1).
̂ Cl
Turbulence Model T∗= 100 T∗= 200 T∗= 300
SL 0.067 0.068 0.075
WALE 0.89 0.81 0.73
TBLE 0.040 0.044 0.041
TBLE_k 0.030 0.038 0.040
SAS 0.14 0.13 0.13
DES_SA 0.104 0.082 0.093
k-ε 0.12 0.12 0.12
Once again, the variation with sampling time is much less than the differences between the values predicted by the different models.
The comparison of these parameters for both the Grids at T∗=100 is shown inTable XI. As expected, there is much more effect of grid refinement on the profile parameters as compared to integral parameters like drag and lift coefficient. The base pressure coefficient shows an increase (in terms of magnitude) for the TBLE models
TABLE V. Comparison of RMS of lift coefficient (Ĉl) with experimental results for Re = 1×106.
̂ Cl
Turbulence Model Grid 1 Grid 2
SL 0.067 0.065
WALE 0.89 0.11
TBLE 0.040 0.031
TBLE_k 0.030 0.035
SAS 0.14 0.58
DES_SA 0.104 0.049
k-ε 0.12 0.10
Schewe15
(Re = 8×105) 0.013
(Re = 1.5×106) 0.033
TABLE VI. Variation of Strouhal Number (St) with Sampling Time for Re = 1×106 (Grid 1).
St
Turbulence Model T∗= 100 T∗= 200 T∗= 300
SL 0.17 0.17 0.17
WALE 0.12 0.12 0.12
TBLE 0.32 0.32 0.32
TBLE_k 0.33 0.33 0.33
SAS 0.26 0.27 0.27
DES_SA 0.16 0.16 0.16
k-ε 0.31 0.31 0.31
which resulted in the increase in drag coefficient. One of the rea- sons for this is the advancement of the angle of separation causing a lowering of the adverse pressure recovery and consequently a higher base pressure coefficient. Despite the change in drag coefficient, the TBLE models still predict the values reasonably close to the experi- mental measurement. The WALE models show a significant delay in separation angle resulting in a huge drop in−Cpb. The adverse pres- sure recovery for the TBLE models was also within the range of the experimental values. The hybrid models show different behavior for grid refinement with the SAS model showing an increase in the base pressure which takes the results even further away from the experi- mental measurements while the DES_SA model shows an improve- ment in the results with grid refinement although the deviation is still high. However, both models show early separation characteristic of laminar separation which indicates the boundary layer turbulence has not been adequately captured. The k-εmodel preforms better than the hybrid models and shows an improvement with grid refine- ment. While the above parameters give an indication of the pressure variation, the complete profiles for the coefficient of pressure (Cp), TABLE VII. Comparison of Strouhal Number (St) with experimental results for Re = 1×106.
St
Turbulence Model Grid 1 Grid 2
SL 0.17 0.24
WALE 0.12 0.32
TBLE 0.32 0.32
TBLE_k 0.33 0.32
SAS 0.26 0.24
DES_SA 0.16 0.24
k-ε 0.31 0.34
LES11 0.35
Schewe15
(Re = 8.2×105) 0.28
(Re = 1.5×106) 0.46
James et al23
(Re = 9.2×105) 0.33
(Re = 1.8×106) 0.26
TABLE VIII. Variation of Mean base pressure coefficient (Cpb) with Sampling Time for Re = 1×106(Grid 1).
−Cpb
Turbulence Model T∗= 100 T∗= 200 T∗= 300
SL 0.66 0.68 0.68
WALE 1.61 1.55 1.48
TBLE 0.33 0.34 0.33
TBLE_k 0.40 0.38 0.38
SAS 0.73 0.73 0.73
DES_SA 0.79 0.78 0.79
k-ε 0.53 0.53 0.53
TABLE IX. Variation of Mean adverse pressure recovery (Cpb−Cp,min) with Sampling Time for Re = 1×106(Grid 1).
Cpb−Cp,min
Turbulence Model T∗= 100 T∗= 200 T∗= 300
SL 1.45 1.45 1.45
WALE 0.32 0.35 0.41
TBLE 2.31 2.35 2.34
TBLE_k 2.22 2.22 2.22
SAS 1.51 1.53 1.53
DES_SA 1.19 1.21 1.21
k-ε 1.95 1.95 1.95
on the other hand, shows the variation over the entire cylinder sur- face. The effect of grid refinement on the pressure profiles is shown inFig. 11. As was evident from the values of profile parameters in Table XI, there is an effect of the grid refinement on the pressure profile for all models though the scale of effect varies from model to model. The SL model shows the minimum impact of grid refinement while the WALE model shows the maximum impact. The TBLE based models show lesser impact as compared to the hybrid mod- els with the constantκmodel showing lesser impact among the two.
Among the hybrid models, the SAS model mainly deviates in the TABLE X. Variation of mean angle of separation (θSep) with Sampling Time for Re = 1×106(Grid 1).
θSep(deg)
Turbulence Model T∗= 100 T∗= 200 T∗= 300
SL 92.73 92.73 92.73
WALE 86.82 88.18 86.36
TBLE 107.71 106.36 106.36
TBLE_k 107.27 105.91 105.91
SAS 95.00 95.00 95.00
DES_SA 89.09 89.09 89.09
k-ε 104.54 104.54 104.54
TABLE XI. Comparison of Pressure profile parameters with experimental results for Re = 1×106.
Turbulence −Cpb Cpb−Cp,min θSep(deg) Model Grid 1 Grid 2 Grid 1 Grid 2 Grid 1 Grid 2
SL 0.66 0.71 1.45 1.35 92.73 94.54
WALE 1.61 0.24 0.32 2.43 86.82 118.17
TBLE 0.33 0.40 2.31 2.19 107.71 104.09
TBLE_k 0.40 0.44 2.22 2.16 107.27 101.81
SAS 0.73 0.90 1.51 1.37 95.00 96.36
DES_SA 0.79 0.63 1.19 1.63 89.09 96.82
k-ε 0.53 0.46 1.95 2.07 104.54 104.78
LES11 0.32 - -
Shih et al.18
(Re = 8.6×105) 0.33 2.30 105.02
(Re = 1.23×106) 0.37 2.02 109.15
separated flow region while the DES_SA model shows deviation in both attached flow region as well as separated flow regions. The k- ε model also shows a lowering ofCp,min as well as an increase in pressure recovery with grid refinement.
A comparison of the profiles of the pressure coefficient with the experimental results for Grids 1 and 2 has been shown inFig. 12 andFig. 13respectively. The experimental results of James et al.23at Re = 2.76×106and of Warschauer and Leene24at Re = 1.26×106 and the numerical results of Catalano et al.11 have been plotted
FIG. 11. Effect of grid refinement on mean pressure coefficient around the cylinder at Re = 1×106. __G1 refers to Grid 1 (Dashed line), _G2 refers to Grid 2 (Bold line).
FIG. 12. Variation of mean pressure coefficient around the cylinder at Re = 1
×106(Grid 1).
for comparison. For the coarse grid, none of the turbulence mod- els apart from the TBLE models predict a pressure profile close to the experimental measurements. While the TBLE models predict a slightly lower value ofCp,min, their base pressure coefficient measure- ments are very close to the experimental results of James et al.23The two TBLE models agree with each other quite closely till the point of
FIG. 13. Variation of mean pressure coefficient around the cylinder at Re = 1
×106(Grid 2).
separation where varyingκmodel in general predicts a lower mag- nitude of base pressure coefficient. As the variation inκ, is triggered only by the adverse pressure gradient, it is natural that the two model predict similar profiles in the region of the favorable pressure gradi- ent. The k-εmodel agrees quite well with the experimental results till the point of separation after which it deviates away. While the WALE model shows a profile which is more characteristic of a sub critical regime, the other models show a presence of a hump which indicates the presence of a separation bubble which is characteristic of the flow regime which is still not fully super-critical.1
For the fine grid, the TBLE models show an increase in the base pressure coefficient (magnitude) while the results show bet- ter agreement in the attached flow region. Even in the post sepa- rated region, the values are still within the experimental predictions.
The WALE model shows a much-improved prediction but shows a much more delayed separation leading to a lower base pressure coefficient (magnitude). However, the pressure rise is not as sudden as is seen with the experimental results. The SL model still shows an early separation and very low adverse pressure recovery. The hybrid models show very similar behavior till the point of separation after which the SAS model deviates to a much higher base pressure coefficient (magnitude). The k-εmodel shows good agreement with the TBLE models till the point of separation as was seen with the coarse grid.
While the major contribution to the overall drag coefficient comes from the pressure variation, the wall shear also contributes to the same although to a much lesser extent. The effect of wall shear is captured through the skin friction coefficient (fs) which is defined as follows.
f¯s= ∣¯τw∣/0.5ρU∞2 (13) Though its contribution to drag and lift force is much lesser as compared to fluid pressure, the physical behavior of the flow near the cylinder is understood through skin friction. As compared to the pressure coefficient profile, the skin friction is much more sensitive to near-wall grid refinement. This is understandable as the pressure variation within the boundary layer is much lesser while the wall shear is strongly dependent on the near wall flow resolution. The variation of the skin friction over the cylinder for the two grids is shown inFig. 14. Apart from the WALE model, all the other models show an increase in skin friction with a refined grid. The WALE and the k-εmodels also show a lot of fluctuations in the region where the flow is decelerating with the WALE model especially showing large oscillations. The TBLE models show minimum variation with grid refinement which may be attributed to the methodology of using the y+ criteria for choosing the velocity input to the model instead of directly taking the first off-wall node velocity. The hybrid models also show a significant increase in the skin friction coefficient with grid refinement.
As compared to pressure profiles, there are very few experimen- tal measurements for skin friction profiles for supercritical Re. The skin friction for the two grids has been plotted along with the exper- imental measurements of Achenbach12and the numerical results of Catalano et al.11inFig. 15andFig. 16. Achenbach,12used a short test cylinder (L/D =3.3) which may have a strong influence on results.
The drag coefficient values measured are also much higher than those obtained by others for similar Re flows. Yet as it is the only data available, it has been plotted for reference.
FIG. 14. Effect of grid refinement on mean skin friction coefficient around the cylinder at Re = 1×106. __G1 refers to Grid 1, _G2 refers to Grid 2.
Almost all the models, overpredict the skin friction profiles as compared to the results of Achenbach for the coarse grid (Grid 1).
However, the TBLE models predict the minimum skin-friction coef- ficients in the attached flow region. Most of the models also show early separation which was also reflected in the higher drag coeffi- cient values given by them. For the fine Grid, we notice a signifi- cant drop in the skin friction measurements by the WALE model which now predicts the least skin friction coefficient. It shows an early separation followed by an immediate re-attachment and much
FIG. 15. Variation of mean skin friction coefficient around the cylinder at Re = 1
×106(Grid 1).
FIG. 16. Variation of mean skin friction coefficient around the cylinder at Re = 1
×106(Grid 2).
delayed separation. This leads to the dropping in the prediction of drag coefficient. Even the numerical result of Catalano et al.11tends to overpredict the skin friction as compared to the experimental results.
B. Flow over the cylinder at Re = 6.5×105 and Re = 2.0×106
Having compared the performance of different models for a supercritical Re of 1.0×106, it was found that for both the grids, the results obtained by the TBLE models seem to in range for most of the experimental results. In order to test the performance of the model further, the TBLE models were used to solve for flow at Re on
either side of the sub-critical Re of 1.0×106. The coarser grid (Grid 1) was used to solve for the Re of 6.5×105while the finer grid (Grid 2) was used to solve for the Re of 2.0×106. All the results have been obtained for a sampling time of T∗=100. The variation of the mean drag coefficient, RMS of lift coefficient and Strouhal number for the TBLE models is shown inTable XII.
As seen fromTable XII, for the case of Re =6.5×105, the values for the mean drag coefficient are within the range of experimen- tal results for the variable κcase while the constantκcase tends to overpredict the mean drag. Again there is a considerable scat- ter even among the experimental results varying from 0.16-0.25.
As there are two possible sub-regimes for this Re,1the lower drag values usually correspond to the symmetric bubble regime while the higher values show the onset of super-critical flow. Again, the RMS of lift coefficient is also overpredicted by both the models as compared to the results of Schewe.15The St, on the other hand, is identical for both the models and is in range of the experimental results.
For the case of Re = 2.0×106, the values for the mean drag coef- ficient are lower than most results. As seen in the previous cases, the TBLE with constantκusually predicts a higher drag coefficient than the one with the variableκ. The RMS of the lift coefficient, on the other hand, agrees quite well with the experimental results. As men- tioned earlier, as the vortex shedding in this regime is irregular, there is a significant scatter in the values of St measured and the values of St obtained from the TBLE models are within the range expected for the supercritical regime.
For the case of Re = 6.5×105, the base pressure coefficient is on the higher side of the experimental results which had resulted in an increased mean drag, especially for the constantκmodel. The pres- sure recovery is within the range of the experimental results despite a higher magnitude of base pressure which might be due to a lower value ofCp,min. The angle of separation as well is within the range of the experimental results.
For the case of Re = 2.0×106, the base pressure coefficient agrees quite well with the experimental results for both the mod- els with the constantκmodel predicting a higher value as it was
TABLE XII. Comparison of mean drag coefficient (CD), RMS lift coefficient(̂Cl)and Strouhal Number (St) with experimental results.
CD Ĉl St
Turbulence Model 6.5×105 2.0×106 6.5×105 2.0×106 6.5×105 2.0×106
TBLE 0.22 0.24 0.028 0.029 0.31 0.36
TBLE_k 0.27 0.26 0.029 0.032 0.31 0.35
Shih et al.18
0.18 0.28
NA NA NA NA
(6.0×105) (1.23×106)
0.21 0.30
(8.5×105) (1.5×106) Schewe15
0.22 0.26 0.022 0.033 0.29 0.46
(7.0×105) (1.5×106) (7.0×105) (1.5×106) (7.0×105) (1.5×106)
0.25 0.36 0.013 0.029 0.28 0.48
(8.0×105) (2.0×106) (8.0×105) (2.0×106) (8.0×105) (2.0×106)
James et al23 0.16 0.32
NA NA 0.33 0.26
(9.2×105) (1.88×106) (9.2×105) (1.88×106)
TABLE XIII. Comparison of Pressure profile parameters with experimental results.
Turbulence −Cpb Cpb−Cp,min θSep(deg)
Model 6.5×105 2.0×106 6.5×105 2.0×106 6.5×105 2.0×106
TBLE 0.33 0.36 2.27 2.28 104.98 104.54
TBLE_k 0.38 0.39 2.22 2.23 106.34 104.09
Shih et al.18
0.26 0.36 2.10 2.0 106.98 110.6
(7.0×105) (1.5×106) (7.0×105) (1.5×106) (7.0×105) (1.5×106)
0.21 0.40 2.30 2.07 105.02 105.2
(8.6×105) (2.0×106) (8.6×105) (2.0×106) (8.6×105) (2.0×106)
for the earlier cases. The pressure recovery for both the models is higher than the experimental results which are due to the predic- tion of a lowerCp,min. As the base pressure predicted by the constant κmodel is higher (magnitude), the pressure recovery is lower. The angle of separation predicted by the models is close to the exper- imental results for a similar Re. For the case of 6.5 × 105, both the models show good agreement in the attached flow region till around 70 degrees after which they deviate slightly which results in a lower value ofCp,minand a higher magnitude of base pressure coef- ficient. The TBLE model with variable-κshows better agreement in the post-separation region.
For the case of Re = 2.0×106, the mean pressure coefficient for both the models overlaps till the separation point after which the constantκmodel deviates to give a higher base pressure coeffi- cient (magnitude). As was evident fromTable XIIIboth the models show a lowerCp,minas compared to the experimental results which results in a higher adverse pressure recovery. The variation of the
FIG. 17. Variation of mean pressure coefficient around the cylinder at Re
= 6.5×105.
pressure coefficient over cylinder for the two Re is shown inFig. 17 and Fig. 18. For Re = 6.5× 105, the results have been compared with those of Flachsbart26at Re = 6.7×105and James et al.23 at Re = 9.2×105. For Re = 2.0×106, the results have been compared with those of James et al.23at Re = 2.7×106.
The skin friction coefficient variation for both the TBLE mod- els for Re = 6.5×105is shown inFig. 19. Even here, the models tend to predict a much higher value as compared to that mea- sured by Achenbach.12The constantκmodel shows a reattachment region similar to that observed by Achenbach12but there is no agree- ment in terms of the magnitude of the second peak or the posi- tion of reattachment. The varyingκmodel though does not show any such reattachment and the nature of the profile is similar to a super-critical profile. Both the models show similar behavior in the post-separation region.
The variation of mean skin friction coefficient for Re
= 2.0×106as seen inFig. 20shows a similar variation as was seen
FIG. 18. Variation of mean pressure coefficient around the cylinder at Re
= 2.0×106.
FIG. 19. Variation of skin friction coefficient around the cylinder at Re = 6.5×105.
with the case of Re = 1.0×106. The predictions for the two mod- els overlap each other in both the attached flow as well as in the post-separation regions. While the actual values do not agree with the experimental results, the nature of the curve is similar as in the angular position of the maxima as well as minima are in agreement with the experimental observations.
Having, thus, compared the performance of the TBLE models for three different Re and against other turbulence models for a Re of 1.0×106, the summary and conclusions drawn are elaborated in the next section.
FIG. 20. Variation of skin friction coefficient around the cylinder at Re = 2.0×106.
IV. SUMMARY
After having solved the supercritical flow case at an Re of 1.0×106with different models, and subsequently solving two other cases of Re of 6.5×105and 2.0×106with TBLE models, the per- formance of the TBLE model can be analysed based on i) Accuracy, ii) Grid Sensitivity and iii) Computation time.
As seen from the previous section, the results of the TBLE models compare very well with the experimental data for the Re = 1.0×106as compared to the other turbulence models. In fact, with the coarser grid, the TBLE models were the only ones which were able to predict results close to the experimental results. Among the TBLE models, it was observed that by varyingκ, the drag predic- tions were generally lower as it also brought down the base pressure coefficient (magnitude). However, the results of both the models were within the bands of experimental range and so it is difficult to compare the models based on that alone. While the skin friction coefficient was generally over-predicted by all the models, the TBLE models gave much lesser skin friction for both the grids with the exception of the WALE model for the fine grid. However, the WALE model showed a lot of scatter in the deceleration region. For the Re of 6.5×105, the models tended to over-predict the mean drag and base pressure coefficients but they were still reasonably close to the exper- imental results. The varyingκmodel tended to perform marginally better than the constantκmodel. For the Re of 2.0×106, the results were in agreement for both the models though the mean drag coef- ficient and base pressure coefficients were slightly under-predicted.
Thus, the overall accuracy of the TBLE models is quite satisfactory.
The effect of grid refinement also shows that the TBLE model is least sensitive to changes in near-wall grids. Even for the coarse grids, the accuracy is still reasonable as compared to the other mod- els especially with the varyingκmodel. While less sensitivity to the wall grid is also seen in the SL model, its accuracy is not as much.
The WALE model shows better accuracy for fine grids but is very sensitive to grid changes. This makes the predictions by the WALE model very volatile and thus difficult to rely upon.
It is also important to know how much additional computation cost is spent in calculating the wall shear stress in a TBLE model for the greater accuracy and grid independence it provides. Hence, a comparison has also been done with respect to the computation time taken per time step. The time depends mainly on the grid and not on the Re number as there is good convergence for all the three Re numbers. These have been compared for the case of Re = 1.0×106 by solving on an 18 core parallel shared memory processor (Intel
®
XEON
®
CPU E5-2620 @ 2.10 GHz) inTable XIV.TABLE XIV. Comparison of computation time (wall clock time in sec) for different grids and models forRe= 1.0×106.
Model Grid 1 (sec) Grid 2 (sec)
TBLE (for both variations) 1.325 2.416
SL 1.150 2.261
WALE 1.174 2.698
DES_SA 1.485 2.889
SAS 2.052 4.425
k-ε 1.458 2.929
