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A theoretical approach is developed for solving for the Reynolds stress in turbulent flows, and is validated for canonical flow geometries (flow over a flat plate, rectangular channel flow, and free turbulent jet). The theory is based on the turbulence momentum equation cast in a coordinate frame moving with the mean flow. The formulation leads to an ordinary differential equation for the Reynolds stress, which can either be integrated to provide parameterization in terms of turbulence parameters or can be solved numerically for closure in simple geometries. Results thus far indicate that the good agreement between the current theoretical and experimental/DNS (direct numerical simulation) data is not a fortuitous coincidence, and in the least it works quite well in sensible ways in canonical flow geometries. A closed-form solution for the Reynolds stress is found in terms of the root variables, such as the mean velocity, velocity gradient, turbulence kinetic energy and a viscous term. The form of the solution also provides radically new insight on how the Reynolds stress is generated and distributed.

We present a theoretical development and a solution for the Reynolds stress in turbulence. Turbulence is considered one of the most difficult problems in fluid physics, or some say, physics in general. It is also quite important as many issues of practical concern, such as weather, aerodynamics, combustion flows and many industrial processes depend on turbulence, and much work has been done on finding some adequate approximations so that immediate problems of turbulent flows can be solved (we do not attempt to list the vast literature in this area). As finding the entire absolute (mean + fluctuations) velocity field is quite difficult, or as some argue an overflow of information, here we focus on finding the Reynolds stress as a function of the “root” turbulence parameters, such as the mean velocity and its gradient, turbulence kinetic energy, in particular its longitudinal component, and also a viscous term. This has been the goal of turbulence theory and modeling: parameterizing the Reynolds stress in terms of known or readily available variables, or so-called the closure problem.

The current theoretical framework leads to the “integral formula” explicitly giving the Reynolds stress in terms of the “root”, calculatable turbulence parameters. In Reynolds-averaged Navier-Stokes (RANS) equation, non-linear terms involving turbulent fluctuation velocities arise since the absolute velocity is decomposed into mean (U) and fluctuating (

In Equation (1), t, x, and y are the time and coordinates, while

If the time mean of the fluctuating velocity does not vary appreciably in time, then we can write a “steady-state” momentum equation, and solve for the gradient of the Reynolds stress.

In conventional calculations, the x-derivatives would have been set to zero for fully-developed flows, and we would be left with a triviality. However, we note that Equations (1) and (2) have been written for a control volume which is moving along with the mean flow velocity, as shown in

The mean velocity, U, appears as a multiplicative factor in Equation (3). C_{1} is a constant that depends on the Reynolds number. Similarly, the gradient of the pressure fluctuation will not be zero in general. However, this term is expected to be significant only for compressible flows, so we omit this term from further analysis in this phase of the work. In Equation (2), we now have a simple integrable expression to find the Reynolds stress, after using Equation (3). If we integrate by parts, we obtain:

For axi-symmetric flow, the results are similar, leading to the following expression for the Reynolds stress.

_{θ}) ranged from 1430 to 31,000 [

the measured Reynolds stress as in

_{τ} = 110 - 650, where Re_{τ} is the Reynolds number based on the friction velocity and channel half-width. The entire data set from the DNS is available on their website [

the mean velocity, turbulent fluctuating velocity components, and various moments of their products. We input the necessary root turbulence parameters into Equation (4), and compare with the Reynolds stress from DNS. The agreement is nearly perfect at low Reynolds numbers in

There is an interesting departure at higher Reynolds numbers, as the solution starts to overshoot the DNS data as y approaches the centerline. The y location where this departure starts to occur decreases (further away from the centerline) at higher Reynolds numbers. This departure is due to the fact that the symmetry boundary condition for channel flows, at the centerline, has not yet been imposed. As noted earlier, the integral term is a “displacement” term accumulating from the wall, and at the centerline the displacement must cancel out. For example, the integral formula (Equation (4)) and its preceding transport equation (Equation (3)) have been derived for flows bounded on one side, such as the flow over a flat plate, and the solution proceeds from y = 0 (wall) onward. For channel flows, the flow is bounded on both sides, leading to the requisite symmetry condition at the centerline. One way to impose the symmetry boundary condition is to force the constant C_{1} to be proportional to the velocity gradient. For example,

With m = 1/3, indeed the calculated Reynolds stress tracks the DNS data fairly well at the Reynolds number of 400, as shown in

We have found a method to derive an expression for the Reynolds stress in simple flow geometries, leading to an “integral formula”. This formula, and the method, works quite well in determining the Reynolds stress based on inputs of root turbulence parameters, such as streamwise component of the turbulence kinetic energy, the mean velocity and its gradient. The predicted Reynolds stress is in good agreement with experimental and DNS data. In particular, if the data are continuous and aligned, then agreement is nearly perfect. There are some nuances and corrections that need to be examined, such as applying symmetry boundary conditions and relating the displacement effect to the flow geometry and Reynolds number. Thus far, the theory has been tested against relatively simple geometries. Would this method be useful in full three- dimensional turbulent flows? That is a question that is being thought of at this time. The fact that u'v' is related to u'^{2} is easier to implement in computational applications as the turbulent kinetic energy can be related to u'^{2}, assuming isotropy, or an equation for u'^{2} can be numerically solved in conjunction with Equation (4). For simple flows, the displacement effect could be effectively treated with Equation (6). Extensions to fully three-dimensional flows will require this displacement effect to be parameterized,

which may not be a simple matter. On the other hand, it was considered difficult to parameterize the Reynolds stress even in simple flows, for quite some time.

This work was conceived immediately after some conversations with friend (s) in Brno, Czech Republic, on Cesky Drahy train which allows for the window to be opened at turbulent relative airspeeds.

Lee, T. (2016) A Theoretical Solution for the Reynolds Stress: Validations in Canonical Flow Geometries. Open Journal of Fluid Dynamics, 6, 272- 278. http://dx.doi.org/10.4236/ojfd.2016.64021