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Modeling of the Turbulent Poiseuille–Couette Flow in a Flat Channel by Asymptotic Methods

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Abstract

Developed turbulent flow of a viscous incompressible fluid in a channel of small width at high Reynolds numbers is considered. The instantaneous flow velocity is represented as the sum of a stationary component and small perturbations, which are generally different from the traditional averaged velocity and fluctuations. The study is restricted to the search for and consideration of stationary solution components. To analyze the problem, an asymptotic multiscale method is applied to the Navier–Stokes equations, rather than to the RANS equations. As a result, a steady flow in the channel is found and investigated without using any closure hypotheses. The basic phenomenon in the Poiseuille flow turns out to be a self-induced fluid flow from the channel center to the walls, which ensures that kinetic energy is transferred from the maximum-velocity zone to the turbulence generation zone near the walls, although the total averaged normal velocity is, of course, zero. The stationary solutions for the normal and streamwise velocities turn out to be viscous over the entire width of the channel, which confirms the well-known physical concept of large-scale “turbulent viscosity.”

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ACKNOWLEDGMENTS

The author is grateful to A.R. Gorbushin and I.I. Lipatov for helpful discussions in the course of this study.

Funding

This work was performed at the Moscow Institute of Physics and Technology and was supported by the Russian Science Foundation, project no. 20-11-20006. Computer support was provided by the Central Aerohydrodynamic Institute and the Federal Research Center “Computer Science and Control” of the Russian Academy of Sciences.

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Correspondence to V. B. Zametaev.

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Translated by I. Ruzanova

APPENDIX

APPENDIX

The problem for second-order perturbations has the form

$$\begin{gathered} O\left( {\frac{\delta }{{{{\delta }^{2}}}}} \right)\,:\quad {{\nabla }^{2}}{{p}_{2}} + 2\frac{{\partial {{u}_{0}}}}{{\partial {{y}_{1}}}}\frac{{\partial {{{v}}_{2}}}}{{\partial {{x}_{1}}}} = - 2\frac{{\partial {{u}_{1}}}}{{\partial {{y}_{1}}}}\frac{{\partial {{{v}}_{1}}}}{{\partial {{x}_{1}}}} - 2{{\left( {\frac{{\partial {{{v}}_{1}}}}{{\partial {{y}_{1}}}}} \right)}^{2}} - 2\frac{{\partial {{{v}}_{1}}}}{{\partial {{y}_{1}}}}\frac{{\partial {{w}_{1}}}}{{\partial {{z}_{1}}}} - 2{{\left( {\frac{{\partial {{w}_{1}}}}{{\partial {{z}_{1}}}}} \right)}^{2}} - 2\frac{{\partial {{u}_{1}}}}{{\partial {{z}_{1}}}}\frac{{\partial {{w}_{1}}}}{{\partial {{x}_{1}}}} - 2\frac{{\partial {{w}_{1}}}}{{\partial {{y}_{1}}}}\frac{{\partial {{{v}}_{1}}}}{{\partial {{z}_{1}}}}, \\ O\left( {\frac{\delta }{\delta }} \right)\,:\quad \frac{{\partial {{{v}}_{2}}}}{{\partial {{t}_{1}}}} + {{u}_{0}}\frac{{\partial {{{v}}_{2}}}}{{\partial {{x}_{1}}}} + \frac{{\partial {{p}_{2}}}}{{\partial {{y}_{1}}}} = - {{u}_{1}}\frac{{\partial {{{v}}_{1}}}}{{\partial {{x}_{1}}}} - {{{v}}_{1}}\frac{{\partial {{{v}}_{1}}}}{{\partial {{y}_{1}}}} - {{w}_{1}}\frac{{\partial {{{v}}_{1}}}}{{\partial {{z}_{1}}}} + \frac{1}{{\operatorname{Re} {{\delta }^{{3/2}}}}}{{\nabla }^{2}}{{{v}}_{1}}, \\ O\left( {\frac{\delta }{\delta }} \right)\,:\quad \frac{{\partial {{w}_{2}}}}{{\partial {{t}_{1}}}} + {{u}_{0}}\frac{{\partial {{w}_{2}}}}{{\partial {{x}_{1}}}} + \frac{{\partial {{p}_{2}}}}{{\partial {{z}_{1}}}} = - {{u}_{1}}\frac{{\partial {{w}_{1}}}}{{\partial {{x}_{1}}}} - {{{v}}_{1}}\frac{{\partial {{w}_{1}}}}{{\partial {{y}_{1}}}} - {{w}_{1}}\frac{{\partial {{w}_{1}}}}{{\partial {{z}_{1}}}} + \frac{1}{{\operatorname{Re} {{\delta }^{{3/2}}}}}{{\nabla }^{2}}{{w}_{1}}, \\ O\left( {\frac{\delta }{\delta }} \right)\,:\quad \frac{{\partial {{u}_{2}}}}{{\partial {{x}_{1}}}} + \frac{{\partial {{{v}}_{2}}}}{{\partial {{y}_{1}}}} + \frac{{\partial {{w}_{2}}}}{{\partial {{z}_{1}}}} = - \frac{{\partial {{u}_{0}}}}{{\partial x}}. \\ \end{gathered} $$
(A.1)

The linear operators in Eqs. (A.1) coincide with those in (2.6), but inhomogeneous quadratic terms additionally appear. The viscous terms are preserved according to the results of [8], where it was shown that the parameter \({{Z}_{i}} = \operatorname{Re} {{\delta }^{{3/2}}}\) can take large, but finite values in the turbulent domain after the laminar–turbulent transition. Additionally, it will be shown later that the parameter \({{Z}_{i}}\) can be eliminated from the resulting equations by applying an affine transformation.

Substituting (2.7) into system (A.1), we obtain the problem for \({{{v}}_{2}}\) and \({{p}_{2}}\):

$${{\nabla }^{2}}{{p}_{2}} + 2\frac{{\partial {{u}_{0}}}}{{\partial {{y}_{1}}}}\frac{{\partial {{{v}}_{2}}}}{{\partial {{x}_{1}}}} = \underbrace { - 2{{{\left( {\frac{{\partial {{{v}}_{{10}}}}}{{\partial {{y}_{1}}}}} \right)}}^{2}}}_{{\text{slow}}\;{\text{term}}\; = \;G} $$
$$\begin{gathered} \underbrace { - \;2\frac{{\partial {{u}_{1}}}}{{\partial {{y}_{1}}}}\frac{{\partial {{{v}}_{{11}}}}}{{\partial {{x}_{1}}}} - 2\left( {2\frac{{\partial {{{v}}_{{10}}}}}{{\partial {{y}_{1}}}}\frac{{\partial {{{v}}_{{11}}}}}{{\partial {{y}_{1}}}} + {{{\left( {\frac{{\partial {{{v}}_{{11}}}}}{{\partial {{y}_{1}}}}} \right)}}^{2}}} \right) - 2\frac{{\partial {{{v}}_{1}}}}{{\partial {{y}_{1}}}}\frac{{\partial {{w}_{1}}}}{{\partial {{z}_{1}}}} - 2{{{\left( {\frac{{\partial {{w}_{1}}}}{{\partial {{z}_{1}}}}} \right)}}^{2}} - 2\frac{{\partial {{u}_{1}}}}{{\partial {{z}_{1}}}}\frac{{\partial {{w}_{1}}}}{{\partial {{x}_{1}}}} - 2\frac{{\partial {{w}_{1}}}}{{\partial {{y}_{1}}}}\frac{{\partial {{{v}}_{1}}}}{{\partial {{z}_{1}}}}}_{{\text{fast}}\;{\text{term}}} \\ \frac{{\partial {{{v}}_{2}}}}{{\partial {{t}_{1}}}} + {{u}_{0}}\frac{{\partial {{{v}}_{2}}}}{{\partial {{x}_{1}}}} + \frac{{\partial {{p}_{2}}}}{{\partial {{y}_{1}}}} = \underbrace { - {{{v}}_{{10}}}\frac{{\partial {{{v}}_{{10}}}}}{{\partial {{y}_{1}}}} + \frac{1}{{\operatorname{Re} {{\delta }^{{3/2}}}}}\frac{{{{\partial }^{2}}{{{v}}_{{10}}}}}{{\partial {{y}_{1}}^{2}}}}_{{\text{slow}}\;{\text{term}}\; = \;E} \\ \end{gathered} $$
(A.2)
$$\underbrace { - \;{{u}_{1}}\frac{{\partial {{{v}}_{{11}}}}}{{\partial {{x}_{1}}}} - ({{{v}}_{{10}}} + {{{v}}_{{11}}})\frac{{\partial {{{v}}_{{11}}}}}{{\partial {{y}_{1}}}} - {{{v}}_{{11}}}\frac{{\partial {{{v}}_{{10}}}}}{{\partial {{y}_{1}}}} - {{w}_{1}}\frac{{\partial {{{v}}_{1}}}}{{\partial {{z}_{1}}}} + \frac{1}{{\operatorname{Re} {{\delta }^{{3/2}}}}}{{\nabla }^{2}}{{{v}}_{{11}}}}_{{\text{fast}}\;{\text{term}}}.$$

The linear operators depending on fast nonstationary variables are written on the left-hand side of the equations, while the nonstationary fast and stationary slow (G, E) inhomogeneous terms are moved to the right-hand side. The contribution to the stationary inhomogeneous terms is made only by the velocity \({{{v}}_{1}}\); therefore, only \({{{v}}_{1}}\) is decomposed into components in (A.2), when necessary. Since the solution of problem (A.2) is linear, it can be represented in the form of a sum of fast nonstationary and slow stationary particular solutions. A detailed analysis of the nonstationary particular solution can be made if we know the first approximation, but, at this stage, it is sufficient to assume that the fast nonstationary inhomogeneous solution of problem (A.2) does not grow with the spatial coordinate \({{x}_{1}}\). This assumption is typical for the method of multiple scales as applied to deriving an equation for the evolution of fluctuation amplitudes of the slow coordinate, and it agrees with experiments showing that the fluctuation amplitude in the channel does not grow in the direction of the basic flow.

The particular solution of problem (A.2) generated by a stationary slow inhomogeneity has the form

$$\begin{gathered} {{{v}}_{2}} = {{x}_{1}}{{V}_{m}}({{y}_{1}}),\quad {{p}_{2}} = {{p}_{{20}}}({{y}_{1}}), \\ {{V}_{m}} = \frac{{E - p_{{20}}^{'}}}{{{{u}_{0}}}},\quad p_{{20}}^{'} = - u_{0}^{2}\int\limits_{{{y}_{1}}}^{ + \infty } {\left( {\frac{G}{{u_{0}^{2}}} - \frac{{2u_{0}^{'}}}{{u_{0}^{3}}}E} \right)} {\kern 1pt} d\eta . \\ \end{gathered} $$
(A.3)

If the stationary right-hand sides G and E in Eqs. (A.2) are arbitrary, then formulas (A.3) mean that the vertical velocity \({{{v}}_{2}}\) is supplemented with a secular term that grows linearly with \({{x}_{1}}\) (in contrast to the first approximation \({{{v}}_{{10}}}\)). As a result, the validity of the original asymptotic expansion (2.5) of the solution is violated. To eliminate the growing spurious solutions (A.3), we set \({{V}_{m}}({{y}_{1}})\) to zero, which yields a necessary condition for the absence of secular terms (solvability condition):

$$\frac{1}{{\operatorname{Re} {{\delta }^{{3/2}}}}}{v}_{{10}}^{{'''}} - {{{v}}_{{10}}}{v}_{{10}}^{{''}} + {{({v}_{{10}}^{'})}^{2}} = 0,\quad {{{v}}_{{10}}}( - 1) = {{{v}}_{{10}}}(1) = 0.$$
(A.4)

This condition is a usual nonlinear differential equation of the third order for the vertical secondary stationary velocity, and only two obvious boundary conditions can be specified for it, namely, impermeability conditions on the walls. It should be noted that condition (A.4) is independent of the streamwise velocity \({{u}_{0}}\) and the slow variable \(x\) is only a parameter in this problem. Therefore, the secondary perturbed pressure has to satisfy the equation

$$\frac{{\partial {{p}_{{20}}}}}{{\partial {{y}_{1}}}} = - {{{v}}_{{10}}}\frac{{\partial {{{v}}_{{10}}}}}{{\partial {{y}_{1}}}} + \frac{1}{{\operatorname{Re} {{\delta }^{{3/2}}}}}\frac{{{{\partial }^{2}}{{{v}}_{{10}}}}}{{\partial {{y}_{1}}^{2}}}.$$
(A.5)

We have obtained an important result: for the solutions of Eqs. (A.2) to have no stationary secular terms, condition (A.4) must hold. This can be achieved in two ways: by specifying the zero vertical velocity \({{{v}}_{{10}}} = 0\) or by satisfying Eq. (A.4) and finding a nonzero velocity \({{{v}}_{{10}}} \ne 0\). The parameter involving the Reynolds number and the channel width can be eliminated from (A.4) by applying the transformation

$${{{v}}_{{10}}} = \frac{1}{{\operatorname{Re} {{\delta }^{{3/2}}}}}V(x,{{y}_{1}}).$$
(A.6)

Note the nonconventional action of viscosity in this case. Usually, low viscosity acts across the basic flow as in laminar flows. At moderate Reynolds numbers, viscous effects are determined by the Laplacian of the velocity. In the case under consideration, the situation is different, since the basic steady flow is vertical and the Laplacian of the vertical velocity is reduced to the only second derivative in the same direction. As a result, the solution transformation (A.6) shows that the molecular viscosity determines the amplitude of vertical velocity, but does not determine the viscous width. It should be noted that this situation corresponds to the physical concept of turbulent viscosity, since it determines the entire width of the channel as a “viscous” length scale, but no artificial hypotheses are used. It should be emphasized once again that the found vertical flow is completely independent of the streamwise velocity, which is explained by the fundamental change in the properties of the channel flow when it passes to a turbulent state. If the channel flow in the laminar phase is parabolic, while developing small perturbations exhibit elliptic properties, then the parabolicity disappears after the laminar–turbulent transition and the velocity and pressure fluctuations begin to prevail over smooth variations in the averaged quantities. This is expressed in the order of formulating and solving the arising problems. First, we need to solve the fast problem of fluctuation evolution (2.6) in the first approximation, and this problem is the basic one. Then the inhomogeneities are computed in the second approximation (A.2) and the corresponding linear equations are solved, next, in the third approximation, etc. The elimination of possible secular terms at each stage gives rise a slow steady secondary flow and determines the conditions for the fluctuation amplitudes. As a result, the basic steady secondary flow represents self-induced, viscous, and distributed transport of the fluid from the flow core to the channel walls. This process is described by an ordinary differential equation depending only on the vertical coordinate. The property of being self-induced means that the nonzero outflow velocity reduces pressure (A.5) on the channel wall. In turn, this increases the outflow velocity, which additionally increases the pressure drop, etc., until the nonlinear effects balance the solution.

Importantly, the secondary stationary solution may (or may not) coincide somewhere with averaged velocities and pressure measured in experiments. This fact is explained by the type of turbulence generation. In free turbulent layers, the generation is distributed and the fluctuation amplitudes are smaller. As a result, the theoretical secondary streamwise velocity [4, 5] agrees well with experiments. In a turbulent boundary layer [6] and in this work, generation is confined to a narrow near-wall domain, and large fluctuation amplitudes and a large domain of fluctuation influence should be expected.

In the first approximation, the pressure has to be zero \(({{p}_{{10}}}(x) = 0)\), since the external pressure is given, while the perturbed pressure in the second approximation, \({{p}_{{20}}}\), can be written as

$${{p}_{{20}}} = \frac{1}{{{{{(\operatorname{Re} {{\delta }^{{3/2}}})}}^{2}}}}\left( {\frac{{\partial V}}{{\partial {{y}_{1}}}} - \frac{{{{V}^{2}}}}{2} + {\text{const}}} \right).$$
(A.7)

Clearly, in the general case, \({{{v}}_{2}}\) and \({{p}_{2}}\) involve terms depending on the slow variable:

$$\begin{gathered} {{{v}}_{2}} = {{{v}}_{{20}}}(x,{{y}_{1}}) + {{{v}}_{{21}}}({{x}_{1}},{{y}_{1}},{{z}_{1}},{{t}_{1}},x), \\ {{p}_{2}} = {{p}_{{20}}}(x,{{y}_{1}}) + {{p}_{{21}}}({{x}_{1}},{{y}_{1}},{{z}_{1}},{{t}_{1}},x). \\ \end{gathered} $$
(A.8)

The next stage of the investigation is to find the secondary streamwise velocity in the channel against the background of the fluid flowing from the core. In the first approximation, the continuity equation yields the fast perturbation of the streamwise velocity:

$$O\left( {\frac{{{{\delta }^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2}}}}}}{\delta }} \right)\,:\quad \frac{{\partial {{u}_{{11}}}}}{{\partial {{x}_{1}}}} + \frac{{\partial {{{v}}_{{11}}}}}{{\partial {{y}_{1}}}} + \frac{{\partial {{{v}}_{{10}}}}}{{\partial {{y}_{1}}}} + \frac{{\partial {{w}_{{11}}}}}{{\partial {{z}_{1}}}} = 0 \to {{u}_{{11}}} = - \frac{{\partial {{{v}}_{{10}}}}}{{\partial {{y}_{1}}}}{{x}_{1}} + {{u}_{{12}}}.$$
(A.9)

To find the basic velocity \({{u}_{0}}\), we use the standard equation for the streamwise momentum in the Navier–Stokes system. It is not given in (2.1)–(2.4), but has the well-known form

$$\frac{{\partial u}}{{\partial t}} + u\frac{{\partial u}}{{\partial x}} + {v}\frac{{\partial u}}{{\partial y}} + w\frac{{\partial u}}{{\partial z}} = - \frac{{\partial p}}{{\partial x}} + \frac{1}{{\operatorname{Re} }}{{\nabla }^{2}}u.$$
(A.10)

Substituting the asymptotic expansions (2.5) of the solution into (A.10) and sequentially extracting the leading terms from the equation, we derive the following equation for first-order perturbations:

$$O\left( {\frac{{{{\delta }^{{1/2}}}}}{\delta }} \right)\,:\quad \frac{{\partial {{u}_{{12}}}}}{{\partial {{t}_{1}}}} + {{u}_{0}}\frac{{\partial {{u}_{{12}}}}}{{\partial {{x}_{1}}}} = \underbrace {{{u}_{0}}\frac{{\partial {{{v}}_{{10}}}}}{{\partial {{y}_{1}}}} - {{{v}}_{{10}}}\frac{{\partial {{u}_{0}}}}{{\partial {{y}_{1}}}} + \frac{1}{{\operatorname{Re} {{\delta }^{{3/2}}}}}\frac{{{{\partial }^{2}}{{u}_{0}}}}{{\partial y_{1}^{2}}}}_{{\text{slow}}\;{\text{term}}} - \underbrace {{{v}_{{11}}}\frac{{\partial {{u}_{0}}}}{{\partial {{y}_{1}}}} - \frac{{\partial {{p}_{{11}}}}}{{\partial {{x}_{1}}}}}_{{\text{fast}}\;{\text{term}}}.$$
(A.11)

As in the equation for vertical fluctuations, the left-hand side represents an operator depending on the fast variables, while the right-hand side consists of stationary slow and nonstationary fast inhomogeneous terms. In formula (A.11), the viscous terms are preserved according to the results of [8], which show that the parameter \({{Z}_{i}} = \operatorname{Re} {{\delta }^{{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-0em} 2}}}}\) takes finite values in the turbulent flow region after the laminar–turbulent transition. Moreover, it follows from (A.6) that this parameter determines only the amplitude of the outflow velocity, but does not affect the behavior of the solution, so we can consider very large values of this parameter, which does not lead to the degeneration of the problem. The streamwise velocity fluctuation \({{u}_{{12}}}\) cannot contain a stationary linear term in \({{x}_{1}}\) (since it has been extracted), which is possible only if the slow right-hand side in (A.11) vanishes:

$$\frac{1}{{\operatorname{Re} {{\delta }^{{3/2}}}}}u_{0}^{{''}} - {{{v}}_{{10}}}u_{0}^{'} + {v}_{{10}}^{'}{{u}_{0}} = 0.$$
(A.12)

It should be noted that the parameter \({{Z}_{i}} = \operatorname{Re} {{\delta }^{{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-0em} 2}}}}\) is eliminated from (A.12) on substituting solution (A.6). This equation is linear and has two linearly independent solutions. Undoubtedly, the resulting equation is viscous, but it is associated with a large “turbulent viscous length scale” in contrast to the small viscous length scale in the turbulence generation zone. The boundary conditions for (A.12) depend on the problem under study.

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Zametaev, V.B. Modeling of the Turbulent Poiseuille–Couette Flow in a Flat Channel by Asymptotic Methods. Comput. Math. and Math. Phys. 60, 1528–1538 (2020). https://doi.org/10.1134/S096554252009016X

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