Transfer of a passive additive in a turbulent boundary layer at very large Reynolds numbers

1. Introduction

In a sequence of works of our group (see refs. 1–5), a new model of a turbulent boundary layer at very large Reynolds numbers was constructed and confirmed by comparison with experiment. According to this model, and contrary to previous models (6, 7), the mean velocity distribution u(y) in the basic intermediate layer between the viscous sublayer and the free stream is represented by two sharply separated scaling laws: Here x and y are the longitudinal (along the flow direction) and transverse Cartesian coordinates, u _∗ = (τ/ρ)^1/2 is the dynamic or friction velocity, τ is the shear stress, ρ is the fluid density, and ν is the kinematic viscosity of the fluid. Furthermore, c ₁ ∼ 50 and c ₂ are constants, λ₀ = c ₁ ν/u _∗ is the thickness of the viscous sublayer, λ = c ₂ν/u _∗ is the thickness of the first self-similar layer adjacent to the viscous sublayer, and δ is the boundary layer thickness. The Reynolds number Re is determined for boundary layer flows by a simple procedure described in refs. 1 and 2. We emphasize that it is different from the Reynolds number Re _θ based on the momentum thickness used traditionally in boundary layer theory. Note (ref. 2) that the second self-similar region adjacent to the free stream is revealed only if the free stream turbulence is low. On the other hand, the first self-similar region was revealed in all experiments performed during the last 40 plus years by various authors (see refs. 2, 4, and 5). The exponent β for the zero-pressure-gradient boundary layers is close to 0.2, whereas in a non-zero-pressure-gradient boundary layer the parameter β depends on the dimensionless quantity where ∂_x p is the pressure gradient (5). The coefficient B is equal to u _∗(u _∗/γ)^β F, where F is a dimensionless function of Re and P.

The equation of mass balance of the passive admixture in the developed turbulent boundary layer has the form (see, e.g., ref. 8) here c is the mass concentration of the admixture, and t is the time. The turbulent diffusion coefficient κ_T is determined as follows: We assume (it is, in fact, a strong hypothesis) that in a developed turbulent flow the mechanisms of the turbulent transfer for momentum and passive admixture are identical; therefore, Here ν_T is the turbulent (eddy) viscosity, K(Re, Sc) is a dimensionless function of the Reynolds Re and Schmidt Sc numbers, Sc = ν/κ, and κ is the molecular diffusivity. The eddy viscosity in a turbulent boundary layer is determined as Therefore, according to 1.1, in the core of the turbulent boundary layer Hence Eq. 1.3 assumes the form We will neglect the viscous sublayer because it is thin and in our model it creates no singularity; i.e., we assume λ₀ = 0. The boundary condition at y = δ should be taken as ∂_y c = 0, because the turbulence of the free stream is assumed to be small. At the internal boundary y = λ the condition of the continuity of the concentration c and of the flux should be fulfilled, so that the derivative ∂_y c will be discontinuous, and the condition should hold: In this note we consider three special problems where the admixture is concentrated in the first intermediate sublayer, so that we can assume that Eq. 1.7 takes the form and we replace the condition at y = λ by the condition of zero concentration at y = ∞. This formulation is appropriate for the initial stages of mixing.

2. Steady Admixture Propagation from a Concentrated Source

We assume that the additive is being injected into the layer at a constant rate through a thin slot, perpendicular to the flow direction on the plate, i.e., at the bottom of the boundary layer. The origin of the coordinates is placed in the source slot, which we assume to be infinitesimally thin. In this case the time derivative in Eqs. 1.3 and 1.9 disappears, and Eq. 1.3 takes the form By integration from y = 0 to y = ∞ at x > 0, we obtain the equation of the conservation of the additive flux because the additive flux outside the source is zero both at the bottom (y = 0) and at infinity, i.e., where the admixture concentration vanishes. Note that the conservation law 2.2 is valid also at a later stage when the admixture fills the whole boundary layer.

Eq. 1.9 takes the form Here we transformed the variable y so that dz = y ^α dy, y = (1 + α)^1/1+α z ^1/1+α, and denoted the resulting constant coefficient by The initial and boundary conditions take the form whereas the condition 2.2 takes the form It is easy to show, using standard dimensional analysis, that the solution to the problem 2.3, 2.4, 2.6 is self-similar where the function f(ξ), ξ = z/(σx)^β, satisfies an ordinary differential equation with the boundary conditions .0 The condition 2.6 is reduced to A simple integration of 2.8 gives The constant C is obtained from condition 2.10, by using ref. 9: where Γ is the Euler's Γ-function, and the solution is obtained finally in an explicit form

3. Unsteady Homogeneous Problems

In this section we consider two problems where the formulation and, consequently, the solution is unsteady but homogeneous in the flow direction. We assume that the wall, the bottom of the boundary layer, is covered by a thin layer containing a passive admixture. In this case the basic Eq. 1.7 takes the form where We assume that the initial concentration of the admixture in the boundary layer is zero. In the first problem, we assume that the concentration of the admixture in the thin layer at the wall remains constant and equal to C. For this problem the initial and boundary conditions take the form Dimensional analysis gives easily where the function f(ξ) satisfies the differential equation and boundary conditions An easy integration gives where P(a, x) is the incomplete Γ-function (9).

For the second problem we assume that a certain charge of the passive admixture is concentrated in the thin layer at the wall, and at t > 0 the plate remains isolated. For this problem we have the conditions and where Q is the admixture charge per unit area; the condition 3.9 is obtained by integration of 3.1 from y = 0 to y = ∞.

In this case dimensional analysis shows that the solution can be represented in the form: where the function f satisfies the ordinary differential equation and boundary conditions Simple integration, taking into account condition 3.9, gives Thus the solution takes the form Remember that α = 3/2 ln Re, and χ is also Reynolds-number-dependent, so both solutions 3.7 and 3.14 demonstrate the Reynolds number dependence of the concentration.

4. Conclusion

The solutions obtained are valid for the initial period of the mass transfer process when the admixture is concentrated in the lower part of the turbulent boundary layer. They are obtained in a simple explicit form. We emphasize that these solutions clearly demonstrate Reynolds number dependence. Moreover, the problem formulation for the new model of the turbulent boundary layer, unlike previous models based on the universal logarithmic law does not lead to any singularities.

Because of the linearity of the basic equations, the solutions obtained can be used for the construction of the solutions to more general problems when the admixture is concentrated in the lower layer. At the later stage when the admixture penetrates to the upper intermediate layer, these solutions can be helpful for the construction of the solutions to the general Eq. 1.7 valid at that stage.