Towards understanding the algorithms for solving the Navier-Stokes equations

In this paper, we proceed exploring the case of non-stationary helical flows of the Navier-Stokes equations for incompressible fluids with variable (spatially dependent) coefficient of proportionality between velocity and the curl field of flow. Meanwhile, the system of Navier-Stokes equations (including continuity equation) has been successfully explored previously with respect to the existence of analytical way for presentation of non-stationary helical flows of the aforementioned type. The main motivation of the current research is the exploring the stability of previously obtained helical flows. Conditions for the stability criteria of the exact solution for the aforementioned type of flows are obtained, for which non-stationary helical flow with invariant Bernoulli-function is considered. As it has been formulated before, the spatial part of the pressure field of the fluid flow should be determined via Bernoulli-function, if components of the velocity of the flow are already obtained.

Note on the stability criteria for a new type of helical flows

Burmasheva N. V., Prosviryakov E. Yu. Isothermal layered flows of a viscous incompressible fluid with spatial acceleration in the case of three Coriolis parameters // Diagnostics, Resource and Mechanics of materials and structures. – 2020. – Iss. 3. – P. 29-46. – DOI: 10.17804/2410-9908.2020.3.029-046. Received: 30.04.2020 Revised: 01.06.2020 Accepted: 19.06.2020 DOI: 10.17804/2410-9908.2020.3.029-046

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Isothermal Layered Flows of a Viscous Incompressible Fluid with Spatial Acceleration in the Case of Three Coriolis Parameters

On a new type of non-stationary helical flows for incompressible 3D Navier-Stokes equations

The principle of pressure maximum is proved for a stationary three-dimensional vortex ideal gas flow (without the assumption of barotropicity). Based on the fact that in regions where the solution is modeled with a high degree of accuracy within the boundary value problem for the Euler equations, the consequences of the Euler equations must also hold, and the obtained subsonic principle is proposed to be used for verification of the numerical solutions of the boundary value problems for Euler equations for an ideal gas and for the Navier-Stokes equations for viscous gas. The conditions of the maximum principle include the value of the Q-parameter, whose surface level image is currently widely used to visualize the flow pattern. The proposed principle of the maximum pressure reveals the meaning of the surface Q = 0. It divides the flow region into the subdomain Q > 0 in which there can be no local pressure maximum and subdomain Q < 0 in which there can be no local pressure minimum. A similar meaning of parameter Q was known for incompressible fluid (H. Rowland, 1880; G. Hamel, 1936). The expression for the Q-parameter contains only the first derivatives of the velocity components, which allows determining the sign (+/–) of Q even for numerical solutions obtained by the low-order methods. An example of the numerical solution’s verification using the subsonic principle of the pressure maximum is presented. Analysis of the results of the numerical calculation of the flow around a moving aircraft carrier in the presence of atmospheric winds showed that if the calculation results are used for the simulation of complex flight modes and analyze the state of the atmosphere from the point of view of safe air traffic, visualizing the flow pattern by Q = const surfaces is not informative. In particular, these surfaces do not reflect the true picture of the wind shear perceived by the aircraft directly entering it. To verify the numerical method, it is sufficient to provide only a surface Q = 0 which has a clear physical meaning.

The Verification of the Calculation of Stationary Subsonic Flows and the Presentation of Results

An expression is obtained for the integral of the equations of plane-parallel viscous incompressible flow expressing conservation of the Bernoulli function along a certain family of lines which, as the viscosity tends to zero, go over into the streamlines. These lines also determine the direction of transfer of the vorticity of the viscous flow.

Some general properties of plane-parallel viscous flows

A new solution for the Navier–Stokes equations has been found for a plane steady-state shear flow of a viscous gas in the gravity field between two vertical walls. For the temperature dependence of the viscosity factor, Sutherland’s formula is accepted; for the heat conductivity factor, two formulas with the same accuracy are used: a known formula for low temperatures (170–1000 K) and a first-proposed formula for high temperatures (800–1500 K). In each of these temperature ranges, a general solution for the Navier–Stokes equations has been obtained. The solution is expressed in terms of a function which satisfies an ordinary second order differential equation having different forms for low and high temperatures depending on the chosen formula for heat conductivity. For low temperatures, the solution of this equation is obtained numerically; for high temperatures, owing to using the new formula for heat conductivity, analytically. Examples of exact solutions are presented.

Viscous Gas Flow between Vertical Walls

The integral invariant of the equations of motion of a viscous gas

Viscous gas flow between vertical walls

For the general case of a spatial isoenergetic flow of ideal gas, Helmholtz’s theorems are generalized and the speed with which vortex tubes move is found, keeping the intensity. It is shown that along the streamline without stagnation point, vorticity either is equal to zero everywhere, or it is non zero at all. The pattern of vortex lines behind the three-dimensional detached bow shock wave is specified.

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Valerii Nikolaevich Golubkin

Papers

Some general properties of plane-parallel viscous flows

Viscous Gas Flow between Vertical Walls

The integral invariant of the equations of motion of a viscous gas

Viscous gas flow between vertical walls

On the vorticity behind 3-D detached bow shock wave