CC BY
0Russian Journal of Nonlinear Dynamics, 2023, vol. 19, no. 2, pp. 219-226. Full-texts are available at http://nd.ics.org.ru DOI: 10.20537/nd230601
NONLINEAR PHYSICS AND MECHANICS
MSC 2010: 76D05, 76R05
Plane-Parallel Advective Flow in a Horizontal Layer of Incompressible Permeable Fluid
K. G. Shvarts
In this paper a new exact solution of the Navier- Stokes equations in the Boussinesq approximation describing advective flow in a horizontal liquid layer with free boundaries, where the vertical velocity component is a constant value, is obtained. The temperature is linear along the boundaries of the layer. Solutions of this kind are used to close three-dimensional equations averaged across the layer in the derivation of two-dimensional models of nonisothermal large-scale flows in a thin layer of liquid or incompressible gas. The properties of advective flow at different values of Reynolds number and Prandtl number are investigated.
Keywords: advective flow, exact solution, Navier-Stokes equation
1. Introduction
Peculiar flows occur in a flat horizontal fluid layer due to horizontal convection. In monograph [5] they are called advective flows to distinguish them from flows arising from classical (vertical) convection. In recent years analytical solutions for Navier-Stokes equations in Boussinesq approximation describing advective flows with complicating factors have been formulated. These include, for example, flows with an internal linear heat source [13], advective flows in the presence of a Navier slip condition [12], in the presence of an acoustic wave [7] and with a weak layer slope [10, 11]. A new class of exact solutions of the Navier-Stokes equations describing advective flows is formulated when the temperature at the layer boundaries is distributed according to the quadratic law. The problem is reduced to solving a nonlinear system of non-stationary one-dimensional equations [3, 4, 6]. On the basis of this theory, an analytical solution describing isothermal flow in a thin horizontal layer of permeable fluid [9] in the presence of a constant vertical velocity at the layer boundaries is presented. A new class of exact solutions of the Navier-Stokes equations in the presence of horizontal convection is constructed [2]. An
Received September 23, 2022 Accepted May 23, 2023
Konstantin G. Shvarts
kosch@psu.ru
Perm State University
ul. Bukireva 15, Perm, 614990 Russia
experimental and numerical study on the structure and stability of a solutal advective flow in a horizontal shallow layer is presented [8].
There is a class of exact solutions used to create new two-dimensional mathematical models describing advective transport in the atmosphere, ocean and a number of technological processes [14]. In particular, a new quasi-dimensional model of admixture propagation from a powerful thermal source (using oil combustion as an example) was constructed based on a three-dimensional convective model, taking into account the inhomogeneity of turbulent diffusion above and outside the source and the constancy of the speed of the vertical hot air jet flowing out of the hydrocarbon combustion source. To solve the two-dimensional equations of motion and heat conduction, we use exact solutions of the original problem describing a flow homogeneous in horizontal coordinates.
In this paper we investigate a new exact solution of the Navier-Stokes equations in the Boussinesq approximation describing the advective fluid flow.
2. Exact solution
Let us consider a thin infinite horizontal layer of an incompressible fluid with flat boundaries 2 = ±h, on which a linear temperature distribution is given along the axis Ox. The boundaries are free, on them a constant nonzero value of the vertical velocity is set (Fig. 1).
zi\
dv" = = 0 ,vz = w0,T = Ax
dz
^ = ïf = 0> vz = w0. T=Ax
Fig. 1. Schematic representation of the problem statement
Choosing as units of length x, y, z, time t, velocity v = (vx, vy, vz), temperature T and
pressure P, respectively, the layer half-thickness h, 9l3j^h , Ah, p0g/3Ah.2 (here v is kinematic viscosity, / is coefficient of thermal expansion, p0 is the average density, A is the constant horizontal temperature gradient at the layer boundary), we obtain the initial Navier-Stokes equations in the Boussinesq approximation in dimensionless form:
dv ->
— + Gr{vVv) = -VP + Vv + Tiz,
divv = 0,
dT 1
— + GrvVT = — AT, dt Pr
(2.1) (2.2) (2.3)
where Gr = is the Grashof number, Pr' = ^ is the Prandt.l number, x is the coefficient of
thermal conductivity, and the Laplace operator A = ^ + + J^-.
At the boundaries of the layer at Here r = ^^ is the vertical Reynolds number. The closed flow condition is imposed
•'x -1
vx dz = 0. (2.5)
A stationary advective flow homogeneous along the horizontal x coordinate is formed in a flat layer of incompressible fluid. It is described analytically
r
vx = u0(z), vy = 0, vz = T = x + 7-0(2), P = Po(x, z). (2.6)
Substituting Eq. (2.6) into the system of equations (2.1)-(2.5), we obtain a boundary-value problem for the system of ordinary differential equations to find the velocity u0(z) and temperature t0 (z):
u(,"(z) - rv'°0(z) = 1,
To (z) — aTQ(z) = Rau0(z),
1 (2.7)
u'0(±1) =0, J u0(z) dz = 0, t0(±1) = 0, -1
where a = rPr is the vertical Pecle number and Ra = PrGr is the Rayleigh number. The general solution of the differential equation for velocity is
, . (C,C.,r. 1\ r2z2 + 2rz(ClZ + 1) + 2(C> + C2r2 + 1)
= + — + C3 + 3 exp(rz)--±-—3-i-. (2.8)
rp2 r / 2r3
Constants C1, C2 and C3 are found from the three boundary conditions for velocity (2.7) and the problem has an analytical solution:
1/1 1 \ 1 / cosh r z2 exp(rz) \
uQ(z) = ----0 - - —+---^-V^ • (2.9)
r \ 6 r2 J r \ smh r 2 r smh r )
The general solution of the differential equation for temperature is
r0(z) = Ra
exp(az) J (/ u0(z) dz^j exp(—az) dz — — + C5 exp(az)
(2.10)
The constants C4 and C5 are found from the two boundary conditions for temperature (2.7) and the analytical solution is:
70(z) = Ra[-/i + /2 + /3 + /4], (2.11)
1
where
l/l 1 \ f z cosh a exp(az)^ J i = Z 7 ~ ~2 7 +
r V6 r2 ) \a a sinh a a sinha /
1 coshr ( az2 + 2z — a cosh a exp(az) \ r sinh r \ 2a,2 a,2 sinh a a,2 sinh a J'
1 / a2z3 + 3az2 + 6z — 3a ^ a2 + 6 cosh a a2 + 6 exp(az) \ 3 r \ 6a3 6a3 sinh a 6a3 sinh a /'
1 /exp(rz) coshr cosh a exp(az)\ r3(a, — r) \ sinhr sinhr sinh a sinh a J
For Pr = 1 (a = r) /4 changes and /4 = £ + (ff^)2 - 1 - ^^
If r = 0, the solution is simplified to:
M*) = ~ (2-12)
To(z) = fg - 10z3 + 9z). (2.13)
Formulas (2.12) and (2.13) are derived from formulas (2.9) and (2.11) as a result of their limit transition at r tending to zero with multiple use of Lopital's rule.
3. Properties of the exact solution
From (2.12) and (2.13) it can be seen that at r = 0 the velocity has an odd cubic profile and the temperature is described by a fifth-degree polynomial. The liquid moves from right to left in the upper half of the layer and from left to right in the lower half. As the vertical Reynolds number increases, the antisymmetric profile of velocity is broken (Fig. 2), and its value decreases. We denote the first component of velocity at r < 0 as u-(z), and at r > 0 as u+(z). Calculations show that u+(z) = -u-(—z). The dependence of r0(z) on the sign of the Reynolds number is similar.
The derivative of the velocity has the following form for r = 0:
z2 - 1
<{*) = -2-- (3-1)
The maximum and minimum values u0(z) are reached at the horizontal boundaries of the layer.
At all r > 0, the module of the minimum velocity exceeds its maximum (Fig. 3a), the maximum value decreases monotonously and the minimum velocity reaches the local extre-mum -0.33876 at r = 0.49 and then also decreases monotonously with increasing r. The value of the z-coordinate, in which the velocity is zero, asymptotically moves to the upper half of the layer (Fig. 3b) and practically does not change (z = 0.155) at r ^ 69.
The Rayleigh number is a scale multiplier in formulas (2.11) and (2.13), so without loss of generality we will further consider Ra = 1. At r = 0, the temperature r0(z) is antisymmetric. It is positive in the upper half of the layer (the heat flux moves along the layer from left to right) and in the lower half it is negative (the heat flux moves along the layer from right to left). The temperature derivative has the following form:
T^) = ^(5Z4-30z2 + 9). (3.2)
The maximum (minimum) value is equal to ±0.021689. It is reached at z = (Fig. 4).
z 1
-0.5
-0.4 -0.2
0.2 0.4
u0(z)
Fig. 2. Velocity profile for different values of the vertical Reynolds number r extreme 0.4 —i----- z 0.15
0.05
-0.4
468 10 0246
r r
(a) (b)
Fig. 3. Dependence of maximum 1 and minimum 2 (a); zero velocity value on Reynolds number r (b)
In the presence of the vertical velocity component at the layer boundaries, the profile of t0(z) changes. At a small value of Prandtl number (Pr = 0.1), the maximum value t0(z) decreases (Fig. 5b) with increasing vertical Reynolds number, and the module of the minimum value decreases at r > 0.49. The heat flux moving in the upper half of the layer from left to right also disappears (Fig. 5a) and at r ^ 441.3 T0(z) ^ 0. The temperature maximum and minimum, as well as the zero value of T0(z), located at r = 0 in the middle of the layer, shift to the upper half of the horizontal liquid layer (Fig. 5c).
At Pr = 1, the maximum value of t0 (z) also decreases (Fig. 6b) with increasing vertical Reynolds number, and the modulus of the minimum value decreases at r > 1.15, which is greater than at Pr = 0.1. The heat flow moving in the upper half of the layer from left to right also disappears and at r ^ 41.1 t0(z) ^ 0 (Fig. 6a). The temperature maximum and minimum as
To(*0
Fig. 4. Profile of t0(z) at r = 0
(a) (b) (c)
Fig. 5. Profile of t0(z) (a), its dependence of maximum 1 and minimum 2 (b), coordinates of maximum 3, zero value 4 and minimum 5 (c) at different values of r for Pr = 0.1
well as the zero-point of t0(z) located at r = 0 in the middle of the layer are shifted to the upper half of the horizontal liquid layer (Fig. 6c). As r ^^ the minimum point tends to z = 1.4.
At Pr = 6.7, the maximum value of t0(z) decreases (Fig. 7b) as the vertical Reynolds number increases, and the modulus of the minimum value begins to decrease at r > 0.17. The heat flow moving in the upper half of the layer from left to right also disappears and t0(z) ^ 0 at r ^ 7.7 (Fig. 7a). The temperature maximum and minimum, as well as the zero value of t0(z), located at r = 0 in the middle of the layer, are shifted to the upper half of the horizontal liquid layer (Fig. 7c). At r > 7.7, the minimum point is reached at about z = 1.4.
4. Conclusion
The presented new exact solution of the Navier-Stokes equations in the Boussinesq approximation describes advective fluid flow in a flat horizontal layer with free boundaries. At the
(a) (b) (c)
Fig. 6. Profile of r0(z) (a), its dependence of maximum 1 and minimum 2 (b), coordinates of maximum 3, zero value 4 and minimum 5 (c) for different values of r for Pr = 1
(a) (b) (c)
Fig. 7. Profile of r0(z) (a), its dependence of maximum 1 and minimum 2 (b), coordinates of maximum 3, zero value 4 and minimum 5 (c) for different values of r for Pr = 6.7
boundaries of the layer a constant nonzero value of the vertical velocity component vz is given. The presence of the third velocity component changes the antisymmetric character of the first velocity and temperature components. As the vertical Reynolds number increases, the velocity decreases, its maximum shifts to the upper half of the layer, and its location in the velocity profile does not change if r is large enough. For all values of the Prandtl number, as the Reynolds number increases, the temperature r0(z) decreases and becomes negative outside the boundaries of the layer, and the higher the value of Pr, the faster the temperature decreases and becomes negative.
Conflict of interest
The author declares that he has no conflict of interest.
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