Solving Equations

UDCУДК 517.927 : [532.511 + 539.366.074.1]/66.074.1

Solution of the Navier–Stokes Differential Equations and Hydroaeroelasticity Problems for the Processes of Inertial Gas‑Dynamic Separation in the Plane Curvilinear Channels

V.I. Sklabinskyi, O.O. Liaposchenko, I.V. Pavlenko, M.N. Demianenko

Sumy State University, Ukraine

In this paper as a result of the analytical solution of the Navier-Stokes equations for gas flow in the plane semicircular annular channel the radial and circumference velocity components were determined taking into account boundary conditions, limitations and hypotheses. Equation for gas leakages determination and expression for pressure distribution were received.

Key words:hHydrodynamics, the Navier–Stokes equations, hydroaeroelasticity, the flow in the plane curvilinear channels, inertial gas -dynamic separation

1. Introduction

The processes of formation and , as well as the separation of the non-uniform heterogeneous dispersion system (emulsions, suspensions, aerosols) play an important role in science and technology. In terms of specific energy consumption and efficiency of separation, methods of inertial gas-dynamic and inertial-filtrating separation, which are differ in the ways of formation of the geometrical configuration of the separation channels, and the character of movement and path of flow, are considered to be optimal [1].

Traditionally, the louver nozzle blocks corrugated packing blocks(corrugated vane), withhich has wavy (sine wave) or zigzag form, when the nozzle is formed as a (corner packsing blocks), are set to separators. The first isare widely used in the construction of domestic separators of domestic production, the second -– in the international. In both cases, the scientific problem of hydrodynamic processes modeling aimed to predict separation efficiency, as well as development of reliable engineering design techniques of typical separation device is a topical problem.

It should be noted that the theory of isothermal motion of fluid (or gas flow) is based on system of two main equations of fluid dynamics: continuity equation and Navier-Stokes dynamic equations of fluid motion (Navier-Stokes). Solution for the given system of differential equations is one of the six most important Millenium problems in hydrodynamics, in particular, they include turbulence that is the essence one of the six outstanding “Millennium Prize Problems” (April 2015), for the solution of which Clay Mathematics Institute at the beginning of the XXI century had appointed a premium of US $ 1,000,000 [2]. For getting the prize to prove or disprove the existence and smoothness of the solution in any of the two variant is enough. The first variant of equation considers the whole three-dimensional space with some restrictions on the growth rate of the solution at infinity. The second variant of equation considers on two-dimensional torus with periodic boundary conditions. In 2014, Mukhtarbai Otelbaiev published a work, where he argued that he had given a complete solution to the problem [3]. In recent times, mathematicians and physicists are keenly discussing the main statements of this workproblem [3]. Also One of them is Terence Tao, the winner of the Fields Medal. He has published preprint there is an opinion concerning of the impossibility of solving “Millennium Prize Problems” of this problem with currently existing methods [4]. Proven aAnalytical solutions of the equations are found only in certain special cases, for when the small Reynolds numbers for the problem is small, and simple geometry of the channels is simple (e.g., Poiseuille flow). In other cases the numerical simulation with computational fluid dynamics and finite element analysis is used.

This article studies the mathematical formulation and solution of the problem of modeling the motion gas-liquid flow motion in the plane a curvilineared channel of the separation devices with rigid stationary walls and dynamic baffle elements that is complicated by the need of solving the dual hydroelasticity and aeroelasticity problems.

2. Problem Objectives and the Mathematical Model

All of these the following mathematical formulations as well as organize the system of nonlinear differential equations of the seco2^nd order , which have an analytical solution only in very rare cases, due to of the simple geometry of the channels as mentioned above.

Hereby , the givenin this paper article has been accepted such simplifications and assumptions as the flat courseplane flow along the curved channel (Navier-Stokes equations are compiled infor two-dimensional space and convenience in the polar coordinate system) is examined. It is expected that overthat flows and changgesing ofin velocity and pressure fields in height channel height are insignificantly in comparison withcompared with the similar parameters for through the length of the channel length.; pPressure differencechange over the width of the channel width is also insignificant due to the small channel width., where the Significant changeing of size of the pressure difference that occurstakes place along the channel length is significant, , herewith the curvilineared viscous flow is accompanied by the process of conversion of the mechanical energy of the flow from the potential (the pressure) to the kinetic energy and vice versa.

Isothermal gas flow in the plane semicircular annular channel is considered due to the circumferential pressure gradient (p = p(φ), ∂p/∂r = 0) that is described in the polar coordinates equations ofby the continuity and Navier-Stokes equations:

(1)

where ρ is gas density; ε is turbulent viscosity coefficient according to Boussinesq hypothesis.

For given conditions at the entry inlet and outlet into of the separation channels and at the out from them (an expense of a continuous phase, speedvelocity, pressure and the stream direction of a stream) taking into account viscositywith a point of viscosity,, an optimal geometric shape of the channel that provides the minimum of total pressure loss, is existsprovided. The exact solution of a problem of optimum profiling represents has significant difficulties. Simple aApproximateing methods based on simple physical picture understanding of hydrodynamically expedient distribution of the speeds of gas velocities in the flow core and near the channel walls are used in practice [1, 5]. In this instancecase, simplification of walls profile of the curvilinear channel assuming that curvilinear sites have constants internal r₁ and external r₂ rradiuses is permitted.

3. Solving Equations

The distribution functions of the radial and circumferential speedsvelocities, satisfying the continuity equation, are accepted as infinite modified power series

(2)

where A_i(φ) are functions to be further defined; f_i(r) are linearly independent functions, which satisfy the boundary conditions of non-penetration of gas into the channel wall (f_i(r₁) = f_i(r₂) = 0); and q is the constant leakage; β(r) = 6(r – r₁)(r₂ – r)/δ²
is the distribution function of the circumferential velocity (β(r₁) = β(r₂) = 0, );
δ
is the radial gap. Functions ψ_i = f_i + rdf_i/dr must satisfy the condition , therefore f_i = (r – r₁)ⁱ(r – r₂)ⁱ.

Because of the small radial gap (δ << r) functions A_i are determined from the equations of motion (1), which are averaged in the gap:

(3)

where are flow coefficients κ_1,2, coefficients of convective inertia forces ξ_i_1,2 and turbulent viscosity forces γ_i_1,2:

(4)

The coefficients (4) are equal to zero for i > 3. Particularly for i = 2 the 1^st equation (3) takes the form of the ordinary differential equation

(5)

with constant coefficients θ = qξ₁/(εγ₁), k² = 2ξ₂/ξ₁, σ = q²κ₁/(εγ₁). General solution is

(6)

where λ_k are the roots of the characteristic equation λ³ – θλ² – 3λ + k²θ = 0. Integration constants C_k are determined by the conditions:

1) А^/(0) = 0 –is the condition of absence of the radial velocity in the inlet section;

2) А(0) = 0 –is the hypothesis of initial circumferential velocity profile;

3) –is the condition of the velocity gradient limitation.

Integration of the 2^nd equation (3) allows to define the pressure distribution

(7)

where р₀ is inlet pressure; the resistance coefficients are a₁ = πρκ₂/δ, a₂ ≈ 0,
a₃ = πρκ₁ξ₁/(3εγ₁δ). Gas leakage is the real root of equation , where
Δp is the pressure difference.

Hereby, in consequence of the analytical solution of the gas motion equations, the radial and circumference velocity components were determined. Boundary conditions, limitations and hypotheses were taken into account. Equation for gas leakages determination and expression for pressure distribution were received.

4. Conclusion

⇐ Предыдущая 123 Следующая ⇒