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Disturbances propagation processes are investigated in two-dimensional boundary layers for the case of strong viscous-inviscid interaction. The speed of upstream disturbances propagation as a function of specific heat ratio and Prandtl number is determined. Formula for speed propagation is developed on the basis of characteristics and subcharacteristics analysis corresponding to the gasdynamic wave processes and processes of convection and diffusion.

Disturbances propagation in the boundary layers is associated with the processes of convection and diffusion [

Analysis of disturbances propagation in three-dimensional boundary layers for strong interaction led to determination of appropriate subcharacteristic surfaces [

It is supposed that flat plate is placed in hypersonic flow at zero angle of attack. It is supposed as well that strong viscous-inviscid interaction regime is realyzed [

where

velocity vector components, density, pressure, full enthalpy, dynamic viscosity coefficient next definitions are intorduced:

Corresponding to the strong viscous-inviscid interaction theory disturbed flow region near the plate contains two-subregions: 1―inviscid flow between shock wave and boundary layer exteral edge, 2―viscous flow in the boundary layer.

In the subregion 1 next asymptotic expansions are valid:

Substitution of (2) expansions into the Navier-Stokes equations and limiting procedure (1) leads to the next system of equations

With the next boundary conditions on the shock wave:

And on external edge of the boundary layer

For subsequent analysis it is needed to get expression connecting boundary layer thickness

which is the tangent wedge formula generalization for unsteady regime.

For subregion 2 next expansions are valid:

Substitution (4) into the Navier-Stokes equations and limiting procedure (1) give unsteady boundary layer equations. Next transformation

leads corresponding mathematical problem to the next form

where is supposed that dynamic viscosity coefficient linearly depends on the temperature.

The last boundary condition corresponds to the prescribed base pressure value.

Let us determine characteristic (subcharacteristic) surfaces

After transformation

Boundary problem (5) takes the form

where

Interaction condition connecting pressure distribution and boundary layer thickness may be transformed to get next relation

Derivative in the left part of (8) may be expressed in accordance with the boundary layer thickness

The problem formulation (5) may be used to determine derivatives on the

After some transformations using (7)-(8) relations, the next expression may be obtained

where

Relation determining subcharacteristic surface has then the form

where a is an average speed of sound or velocity of subcharacteristic surface

Expression (10) is in fact known integral Pearson generalization [

The sign of

Relation (10) has simple physical explanation. In hypersonic boundary layer exists average velocity value. If average speed of sound is larger than this average velocity than the flow inside boundary layer is subcritical and disturbances can propagate upstream, otherwise the flow will be supercritical.

Relataion (10) may be deduced from (11) by easy way. If we will use moving coordinate

In the moving coordinate system velocity in the boundary layer equals

As an example we can get dependence of an upstream disturbances propagation on specific heat ratio and on

Prandtl number for self similar boundary layer equations solution.

Velocity and full enthalpy profiles were obtained as a result of the next boundary problem solution

These profiles have been used to obtain upstream disturbances propagation velocity

Dependences of an upstream disturbances propagation velocity

We may conclude that Prandtl number influence on

It was supposed that specific heat ratio changes from unity (for polyatomic gas) up to the value 5/3 (for monoatomic gas). It may be concluded that speed of upstream disturbances propagation is larger for larger

This work was partially supported by the Russian Foundation for Basic Research (project № 13-01-06249).