A Solution Method for the Reynolds-Averaged Navier-Stokes Equation T.-W. Lee Mechanical and Aerospace Engineering, Arizona State University, Tempe, AZ, 8587 Abstract- Navier-Stokes equations are difficult to solve, and also the Reynolds stress that arises in the Reynolds-averaged Navier-Stokes (RANS) equations is associated with the closure problem. We present a unique method for solving for the Reynolds stress in turbulent canonical flows, which leads to a solvable set of equations for the RANS. An iterative solution is demonstrated for channel flows wherein the results agree with the data. Further work remains for more general geometries, but a solution method is found for the first time in an inhomogeneous turbulent flow. *Corresponding author: T.-W. Lee Mechanical and Aerospace Engineering, SEMTE Arizona State University Tempe, AZ 8587-6106 Email: attwl@asu.edu 0
Nomenclature C 1, C = constants Re = Reynolds number based on friction velocity U = mean velocity in the x direction u = fluctuation velocity in the x direction u rms = root-mean square of u u v = Reynolds stress = boundary layer thickness = modified kinematic viscosity m INTRODUCTION In the Reynolds-averaged Navier-Stokes equations (RANS), the Reynolds stress represents the momentum transport by turbulence. Solving the full Navier-Stokes equations, if it were possible, would generate far too much details, and in any event some sort of averaging is required to extract useful information from a very large number (possibly infinite) of realizations. Thus, solving the RANS is of significance. Finding the Reynolds stress in RANS has profound implications in fluid physics, and also is a long-standing unsolved problem in mathematical physics (so-called the closure problem). Many practical flows are turbulent, and require some method of analysis or computations so that the flow process can be understood, predicted and controlled. This necessity led to several generations of turbulence models including a genre that models the Reynolds stress components themselves. The conventional approach of writing higher moments for the Reynolds stress, then attempting to model them, involves ever increasing number of unknown terms and corresponding complexities. Recently, we presented a theoretical basis for determination of the Reynolds stress in canonical flows [1-3]. It is based on the turbulence momentum for a control volume moving at the local mean flow speed. The predicted Reynolds stress is in good agreement with 1
experimental and DNS data [1-3]. In addition, a transport equation for u can be written in a similar manner, which is also validated with experimental and DNS data [4, 5]. In this work, we demonstrate this method for solving the RANS in rectangular channel flows. TURBULENCE TRANSPORT EQUATIONS The framework is based on the momentum and u balance for a control volume moving at the local mean flow speed. u is the turbulence fluctuation velocity squared, or the longitudinal component of the fluid kinetic energy. A concept that leads to the governing equation for the Reynolds stress is the differential transform to convert the streamwise gradient to lateral (Eq. ). The derivation has been presented elsewhere [1-5], but to place this work in context we briefly reiterate the main steps. For a control volume moving at the local mean speed, as shown in Figure 1, the momentum balance can easily be written as: ( u' v') u' x urms ' (1) m The quantities (u v, u, and u rms ) are all Reynolds-averaged. We note that in this framework, the complex coupling of the mean and fluctuating components disappear, and the dreaded higher-order term is the Reynolds stress itself. If one pauses to think about this picture, the Reynolds stress is simply due to the momentum balance between the transport (u ) and viscous terms, just as the mean momentum. This picture is so simple that it is hard to believe that it was not conceived of before, but verifications with various data sets confirm the validity of this approach [1-5]. The gradient of the pressure fluctuation is expected to be significant only for
compressible flows, so we omit the pressure fluctuation term for incompressible turbulence at low Mach numbers [1-5]. In conventional calculations in the absolute coordinate frame, the x- derivatives would have been set to zero for fully-developed flows, and we would be left with a triviality. However, for a boundary-layer flow as an example (Figure 1), the boundary layer grows due to the displacement effect. The mass is displaced due to the fluid slowing down at the wall, as is the momentum, and turbulence parameters as well. The boundary layer thickness grows at a monotonic rate, depending on the Reynolds number. Thus, if one rides with the fluid moving at the mean velocity, one would see a change in the all of the turbulence properties, as illustrated in Figure 1. This displacement effect can be mathematically expressed as: x C1 U () I.e., the fluid parcel will see a different portion of the boundary layer in the y-direction, and how much difference it will see depends on how fast the fluid is moving along in the boundary layer. Thus, the mean velocity, U, appears as a multiplicative factor in Eq.. C 1 is a constant that depends on the Reynolds number. Eqs. 1 and lead to the turbulence momentum equation. ( u' v' ) u' C1U urms ' (3) m 3
Figure 1. Illustration of the momentum balance for a moving control volume (top) and an illustration of the displaced control volume (bottom). 4
In Eq. 3, u = (u rms ). m is the modified viscosity, the value of which is close to but not equal to the laminar viscosity [4]. Eq. 3 would indicate, if proven to be correct, that complex modeling of the Reynolds stress is not necessary, and that we can use the turbulence momentum balance to determine the Reynolds stress from root turbulence parameters [1-5]. Indeed, our work [1-5] thus far and the results below establish that a simple coordinate transformation leads to the capture of the internal dynamics of turbulence momentum transport and therefore the Reynolds stress. In addition, we can write an equation for transport of u, in an analogous manner. ( u' CU 3 ) ( u' v' ) u rms ' (4) m In this formulation, the source and sink terms are simplified to just the viscous dissipation term in Eq. 3. The higher-order terms are resolved in terms of root-mean square variables: u v = (u v )u rms and u 3 = (u rms ) 3. The appearance of u v term in Eq. 4 may bring us back to the closure problem, albeit in a simpler equation. However, recalling that the current formulation is for a control volume moving at the local mean velocity, and in u v we are only writing the lateral transport of u, similar to the fact the mean transport can be written in terms of its components (e.g. ( U V ) ). The approximation u v = (u v )u rms is justified for this moving control volume, and by the results in Figure 3. For fully-developed, rectangular channel flows, the RANS is simplified to: 5
U 1 dp ( u' v' ) (5) x In principle, Eq. 3-5 can be solved simultaneously. In order to demonstrate the general framework and a solution method, we consider an iterative solution for a simple rectangular channel flow, for which much is known about the mean and fluctuation velocities [6-8]. Validity of the transport equations has been verified in our concurrent works [4, 5]. Here, we briefly show an example in rectangular channel flow. We can obtain the left-hand side (LHS) of Eq. 4 from u v in the DNS data for rectangular channel flows [6, 7], by computing its derivative ( u' v' ) numerically. In Iwamoto et al. [6, 7], DNS data for Re up to 590 are presented. We can use this data to compute the terms on the RHS of Eq. 3. Only the data for U and u are needed, since u rms is equal to the square root of u. The derivatives are computed using a second-order method. This comparison is shown in Figure, where the LHS and RHS of Eq. 3 are quite well balanced, in spite of the fact that we are taking first- and second-derivatives of discrete data. Therefore, the conservation of turbulence momentum (Eq. 3) holds for channel flows. This has some implications for analysis and computations of turbulent flows: complex models for the Reynolds stress are not necessary, and there is a simple momentum balance that leads to the Reynolds stress (Eq. 3). 6
Figure. Comparison of the gradient of the Reynolds stress for rectangular channel flows. Lines represent current results (RHS of Eq. 3). Data symbols (DNS data): circle (Re = 110), diamond (150), square (300), triangle (400), + (650) [6]. The transport equation for u (Eq. 4) can also be validated using DNS data, as shown in Figure 3, where the LHS of Eq. 4, 3 ( u' ) C U (plotted with data symbols) is compared with the RHS (lines) for various Reynolds numbers. As noted above, we equate the dreaded higher-order term u v with u v *u, or the Reynolds stress times u rms. This is an approximation that serves to suppress the introduction of another unknown variable v rms. However, comparisons with data (Figures 3) show that this approximation works reasonably well. After all, Eq. 4 only attempts to capture the transport of u, and this is accomplished by u v in the y-direction. The initial peak is closely matched by the RHS of the transport equation at all Reynolds numbers, while the 7
second negative peak is underestimated at higher Reynolds numbers. The locations of these peaks are nearly perfectly picked off by Eq. 4. The underestimation of the second peaks may be due to simplification of the source terms into a single dissipation term, the inaccuracy associated with equating u v with u v *u, or that the turbulent kinetic energy in the x-direction (u ) cannot be individually balanced as in the momentum transport. The viscous dissipation cannot distinguish the kinetic energy components in x- or y-direction. For the time being, we retain these approximations to demonstrate that a set of equations to solve for Reynolds stress can be formulated, and refinements to Eq. 4 will improve the accuracy of the transport equation. Figure 3. Contributing terms to the Reynolds stress gradient, 3 ( u' ), in Eq. 4. Data symbol: ( u' v' ) C U from DNS data.; lines: RHS of Eq. 4. 8
Further verification is shown in Figure 4, for rectangular jets. Using experimental data of Gutmark and Wygnanski [8, 9], we can again evaluate Eqs. 3 and 4. For rectangular jets, agreement is quite good for both Eqs. 3 and 4, due to the fact that the viscous terms are very small and only the transport terms affect the balance in a significant way [4]. There are some discrepancies in the initial slope and away from the centerline, but given the fact that we are taking first- and second-derivatives of experimental data, Figure 4 indicates that correct physics of the turbulence transport is captured by Eqs. 3 and 4. Figure 4. Validation of the transport equations, Eq. 3 (LHS = circles) and Eq. 4 (LHS = triangles) for rectangular jets. Lines are the RHS s of Eq. 3 ( ) and Eq. 4 ( ). The data from Gutmark and Wygnanski [8, 9] have been used. 9
A SOLUTION FOR REYNOLDS-AVERAGED NAVIER-STOKES EQUATIONS IN RECTANGULAR CHANNEL FLOWS As noted above, Eqs. 3-5 contain three unknowns (U, u, and u v ), which in principle can be solved. Here, we demonstrate a simple, iterative solution. We start by assuming a u profile, as a function of y/d. Eqs. 4 and 5 can be numerically integrated to give U and u v. Then, these U and u v profiles are input to Eq. 3 to see if the LHS and RHS are matched. u profile is then corrected until a match within a small error is found. The Reynolds stress obtained in this manner is in very good agreement with the DNS data near the wall, as shown in Figure 5, except of course the errors in Figure for the gradient of the Reynolds stress propagate to the Reynolds itself. They also tend to deviate away from the channel wall at higher Reynolds numbers. This is due to the suppression of the displacement effect for confined flows. The differential transform, Eq., is based on continuous displacement; however, for channel flows this displacement is suppressed at the centerline, causing errors. Corrections to this error has been discussed in our earlier works [1,]. The velocity profiles obtained in this manner also replicate the increasingly steep velocity gradients at the wall, as shown in Figure 6. They compare quite closely with velocity profiles as calculated using the DNS data for u v, through Eq. 5. However, neither velocity profiles reproduce the power-law curves, but this is not due to the errors in u v as computed through Eq. 3, as both the DNS data for u v and current results give essentially identical velocity profiles. The source of the discrepancy between velocity profiles obtained from Eq. 5 and the DNS data evidently resides in the use of Eq. 5, which is a straight-forward RANS for channel flows. Identification of this error is currently being sought. 10
Figure 5. Comparison of the Reynolds stress. Data symbols: circle (Re = 110), diamond (150), square (300), triangle (400), + (650) [6, 7]. Figure 6. Comparison of the velocity profiles. Dotted line is the laminar velocity profile. Data symbols: circle (Re = 110), diamond (150), square (300), triangle (400), + (650) [6, 7]. 11
CONCLUSIONS We have found a method to solve for the inhomogeneous turbulent flows. The method is based on a simple, but novel way of writing the turbulence transport equations. These equations along with RANS provide a set of solvable equations (equal to the number of unknowns). The results agree with the DNS data, and the errors are attributed to the displacement effect and approximations made in u equation (Eq. 4). Both of the sources of error can potentially be remedied, which will improve the accuracy of the solution. Nonetheless, a meaningful solution method for an inhomogeneous turbulent flow has been found for the first time, as demonstrated for rectangular channel flows. REFERENCES [1] Lee, T.-W. and Park, J.E., Further Applications of the Integral Formula for Determination of the Reynolds Stress in Turbulent Flows, 8th AIAA Theoretical Fluid Mechanics Conference, AIAA AVIATION Forum, (AIAA 017-4165), https://doi.org/10.514/6.017-4165 [] Lee, T.-W., and Park, J.E., Integral Formula for Determination of the Reynolds Stress in Canonical Flow Geometries, Progress in Turbulence VII (Eds.: Orlu, R, Talamelli, A, Oberlack, M, and Peinke, J.), pp. 147-15, 017. [3] Lee, T.-W., and Park, J.E., Integral Formula for Determination of the Reynolds Stress in Canonical Flow Geometries, APS November 016 Bulleting of American Physical Society. [4] Lee, T.-W., arxiv e-print, http://arxiv.org/abs/1708.0161. [5] Lee, T.-W., Turbulence Transport Equations and Solving for the Reynolds Stress in Turbulent Flows, APS Fluids Meeting, November 017, submitted. [6] Iwamoto, K., Sasaki, Y., Nobuhide K., Reynolds number effects on wall turbulence: toward effective feedback control, International Journal of Heat and Fluid Flows, 00, 3, 678-689. [7] Iwamoto, K. and Nobuhide Kasagi, http://thtlab.jp/dns/dns_database.htm. [8] Gutmark, E. and I. J. Wygnanski, The Planar Turbulent Jet, Journal of Fluid Mechanics, 1970, Vol. 73, Part 3, pp. 466-495. [9] Index of turbulence data, https://torroja.dmt.upm.es/turbdata/agard/ 1