## Abstract

Some hydrodynamical problems in which viscous–inviscid interaction is important are analysed. These problems are: disturbances propagation in channels, three-dimensional nonlinearly disturbed flows and the blow-off phenomenon.

## 1. Introduction

Two papers among scientific works devoted to investigations of viscous fluid flows played a key role for hydrodynamics development. The first paper belongs to Ludwieg Prandtl (1904) and was presented at the Mathematical Congress in Heidelberg. This work formulated a basis of the boundary-layer theory. The theory was based on the experimental data and physical considerations about the influence of negligible viscosity in flows with high Reynolds numbers. The second paper was later written by Werner Heisenberg who is widely known as a founder of quantum mechanics. His publication (Heisenberg 1926) was devoted to the hydrodynamic stability analysis for a high Reynolds number flow. Later, both scientific directions associated with the boundary-layer analysis along with the stability phenomena were investigated intensively.

More than 50 years ago, Sir James Lighthill (1953) proposed a mathematical model describing propagation of disturbances in laminar boundary layers and formulated a linear mathematical problem where processes of viscous–inviscid interaction played an important role. Subsequent success was associated with the formulation and development of asymptotic methods of mathematical physics (Friedrichs 1953; Kaplun 1954; Lagerstrom 1957) that were appropriate to analyse hydrodynamic problems for large or small values of similarity parameters.

Asymptotic analysis led to the conclusion that viscous–inviscid interaction processes play a key role when boundary-layer separation takes place. To describe this phenomenon it was necessary to develop a nonlinear theory of ‘free interaction’ (Neyland 1969; Stewartson & Williams 1969; Messiter 1970).

The development of hydrodynamic stability theory was associated mostly with the analysis of linear processes, but some results for weak nonlinear processes were obtained as well (Stuart 1971; Landau & Lifschitz 1959).

As a next step, the theory of free interaction was generalized to describe unsteady phenomena (Ryzhov 1977; Zhuk & Ryzhov 1979). The linear variant of this theory can describe long-wave instability in boundary layers. In fact, the results following from this theory for small non-dimensional amplitudes led almost to the same results as were obtained from the analysis of solutions of the Orr–Sommerfeld equation (Heisenberg 1926). This fact may seem trivial, however, asymptotic theories can provide significant progress in the development of nonlinear theory of hydrodynamic stability. Therefore, asymptotic theories have prospects in describing complex flows, such as the nonlinear stage of the laminar–turbulent transition, unsteady boundary-layer separation, laminar sublayers in turbulent flows, and so on. Some examples of such applications are presented in this paper.

## 2. Problem formulation

Let us consider the flow in the laminar boundary layer on a flat plate which is influenced by a pressure disturbance. The corresponding pressure change may be connected with a change in boundary conditions owing to surface distortions for example, or with disturbances in the external inviscid flow and it is assumed that the disturbed region is located at some characteristic distance from the leading edge. It is also assumed that flow parameters in the free stream are used to introduce non-dimensional coordinates and flow functions. The Reynolds number is assumed to be large. Analysing the flow near the wall, we can evaluate the thickness of the region where longitudinal velocity changes are nonlinear. The characteristic thickness of this region, *δ*, depends on disturbances amplitude, for example, on the pressure amplitude Δ*p*≪1, so that *δ*∼*ϵ*Δ*p*^{1/2}, where *ϵ*=Re^{−1/2}. Owing to nonlinear changes in the flow functions, this thickness is comparable to the corresponding change of this region thickness. For supersonic or subsonic flows, this change is much larger than the change of the main part of the boundary-layer flow where the longitudinal velocity is of the order of unity. The pressure induced in the external inviscid flow may be determined in correspondence with the linear theory of inviscid flows Δ*p*∼Δ*δ*/Δ*x*, Δ*x*∼*ϵ*(Δ*p*)^{−1/2}, where Δ*x* is the characteristic longitudinal length of disturbed flow. It can be shown that this length is much larger than the boundary-layer thickness. It can also be shown that for the pressure disturbances with amplitudes larger than *Re*^{−1/4}, the flow in the region where nonlinear changes take place is inviscid in the first approximation (Neyland 1974; Zhuk & Ryzhov 1982; Lipatov & Neyland 1987). Therefore, it is necessary to introduce an additional relatively thin region near the wall where viscous forces are important.

We assume that disturbances may be caused by boundary conditions such as geometry distortions (steady or unsteady), blowing or suction and/or by disturbances originating from external flow (acoustic waves).

After substituting the corresponding asymptotic expansions expressed in similarity parameters into the Navier–Stokes equations, we can obtain the following equations for the leading terms:(2.1)Boundary conditions for the velocity components and pressure disturbance at large distances upstream from the interaction region are determined as a result of matching with the solution for undisturbed shear flow(2.2)The solution of partial differential equation (2.2) can be expressed in the following form:(2.3)Substitution of (2.3) into (2.1) then gives the equation(2.4)In this formulation, the pressure distribution is not known beforehand and can be found from the simultaneous solution of the equations for external flow and flow in the nonlinearly disturbed region. Subsequent analysis depends on the external flow. For supersonic flows we arrive at the Burgers equations, for subsonic flows at the Benjamen–Ono equation.

## 3. Disturbance propagation in channels

Here, we consider supersonic flow in a two-dimensional channel. It is assumed that there is an inviscid irrotational core flow surrounded by relatively thin wall boundary layers. Various regimes of disturbance propagation in channels were analysed by Smith (1976, 1977), Bogdanova & Ryzhov (1983), Ruban & Timoshin (1986) and Nikolaeva & Trigub (1995). In contrast to these papers, we assume that the amplitude of pressure disturbances induced by distortions of the channel surface or by ambient pressure change is relatively large (in comparison with the triple deck scale) namely, *ϵ*^{1/2}≪Δ*p*≪1. It is also assumed that *d*/*l*≥*ϵ*/Δ*p*^{1/2}; this relation means that the core flow contains inviscid flow and that the boundary-layers thickness located on the walls are thinner than the channel width. Another assumption is that the channel width is of the order of longitudinal scale of the region of viscous–inviscid interaction.

The solution in the region where the longitudinal velocity is changed nonlinearly may be written as follows:(3.1)

As a result of substitution of expansions (3.1) into the Navier–Stokes equations with the limiting procedure(3.2)the following equations can be obtained:(3.3)with the boundary conditions(3.4)A similar system can be deduced also for the upper wall. Let us denote by the (−) index solution for the lower wall and by the (+) index the upper wall solution. Then the solution of (3.3) may be presented in the form(3.5)

The similarity parameter determines the distance between a fixed point on one wall and the point reached by reflected disturbance from the other wall. This parameter diminishes if the wall temperature increases while skin friction decreases if the Mach number tends to unity or the amplitude of disturbance diminishes.

From equations (3.3) and (3.5) the following equations can be obtained:(3.6)where . Distributions of the functions A_{3−},A_{3+} induce disturbed pressure in the external flow, which can be found from the solution for the core region where disturbed pressure is described by the wave equation. The appropriate solution which takes into account influence of disturbances coming from both channel walls has the form(3.7)where functions *δ*_{w+} and *δ*_{w−} determine channel wall distortions. Eventually, we arrive at the following system of equations:(3.8)where index 3 is omitted.

It is assumed that at large distances upstream, the pressure disturbances are absent so that the boundary condition(3.9)is valid. At large distances downstream, the conditions(3.10)are satisfied or, alternatively, some non-reflecting boundary condition is imposed if the reflected wave reaches the downstream boundary.

For large values of *Δ*, equation (3.8) may be transformed into two independent Burgers equations describing disturbed flow near upper and lower walls.

The system of equations deduced above describes wave processes in channels with the effects of multiple reflections from the walls along with the disturbances propagation upstream and downstream through boundary layers.

Let us consider first the case of symmetrical change of the wall geometry having the following form:(3.11)

The problem (3.8)–(3.10) was solved numerically with an implicit marching procedure. It was found that when the parameter *Δ* diminishes, various disturbed regions come together developing into a unique disturbed region. The form of solution is then changed and disturbances move downstream as oscillatory packets and also in the form of waves travelling upstream. This is illustrated in figure 1 where results of the pressure calculation are presented for the time moment *t*≈20.

The results presented here show how the similarity parameter influences the process of disturbances propagation in channels. Starting from a solution typical for the Burgers equation (for large values of *Δ*), the solution develops a more complex character with downstream propagating oscillatory waves and wave packets spreading upstream.

Solutions for antisymmetric surface disturbances may be obtained if we take *δ*_{w+}=−*δ*_{w−}. Calculations show then that the obstacle initiates a wave moving downstream. Simultaneously, a reflected wave moves upstream. The behaviour of waves looks in some sense similar to solutions of the *K* d*V* equation. Two waves intersect and continue to move in their original directions without significant change of the amplitude and shape. This evolution is presented in figure 2 where the solid line corresponds to an earlier time moment.

It should be mentioned that the model analysed is valid only during a limited time period. The reason for this is that at the bottom of the inviscid nonlinearly disturbed flow, the boundary layer is located. The solution for the boundary layer may terminate in a singularity. However, up to this moment, equation (3.8) and the solutions computed are still valid.

Preliminary computations proved this conclusion. To investigate a combined influence of viscosity and nonlinear interaction, it is necessary to assume that the pressure amplitude is of the order of Δ*p*∼*Re*^{−1/4} which was not considered in this paper.

## 4. Nonlinearly disturbed three-dimensional flows

For three-dimensional nonlinearly disturbed flows, it is impossible to present the solution in the form of equation (2.3). As a result, solution of the full Euler equations with a degenerated equation for normal momentum (∂*p*/∂*y*=0) is required.

This system of equations may be written as follows:(4.1)equation (4.1) may be transformed as follows:(4.2)where the vorticity components have the form(4.3)

An initial field of vorticity at large distances upstream from the region of interaction may be determined as a result of matching with the solution for undisturbed boundary-layer flow where(4.4)If we define(4.5)then these functions are described by the following system of equations:(4.6)

The pressure distribution should be determined as a result of matching the procedure with a solution for the external inviscid flow. For example, if the external flow is hypersonic and if the interaction parameter is small (*Χ*=*Mδ*≪1), then(4.7)where *Δ* is a total change of the boundary-layer thickness. This is dependent on wall distortion thickness, *δ*_{w}, and on the boundary-layer thickness.

This regime is described by the following system of equations:(4.8)It may be shown that for two-dimensional flows, this system of equations can be transformed to the Burgers equation.

## 5. Boundary layer blow-off

It was shown by Catherall *et al*. (1965) that distributed flow injection through a flat plate can lead to the blow-off phenomenon (a special form of the boundary-layer separation). The solution has a singularity at the point where the skin friction tends to zero. As a result, the boundary-layer equations should be reconsidered to take into account processes of viscous–inviscid interaction. This chapter contains results obtained on the approximate model basis.

It is assumed here that owing to the longitudinal velocity smallness (in some region near the wall forming the main part of the boundary-layer thickness), effects of nonlinearity are negligible. Therefore, in this region viscous effects and the longitudinal pressure gradient are important.

It follows that the corresponding equation for incompressible flow has the form(5.1)The following solutions can be obtained:(5.2)The appropriate solution for the boundary layer with a zero pressure gradient may be written as(5.3)

In the region located over the main region, some intermediate region is assumed where the flow is still linear but the pressure gradient influence continues to be negligible so that(5.4)and then matching with the solution in the wall region leads to(5.5)

After transformation of variables(5.6)the change in the boundary-layer thickness may be expressed as(5.7)where .

We can assume that on the length where viscous–inviscid interaction processes are important, the changes in the boundary-layer thickness are relatively small. Otherwise, owing to flow injection and to subsequent growth of the boundary layer and diminishing shear stress, the processes of viscous–inviscid interaction start to influence the corresponding part of the boundary-layer flow. Then the total boundary-layer thickness may be presented as(5.8)and, after transformations(5.9)we obtain(5.10)

To define the interaction problem, an interaction law should be specified. For a supersonic external flow, the appropriate law is given by Ackeret's formula, while for subsonic flow this law is represented by Hilbert's integral,(5.11)

After transformations(5.12)the following equations are obtained:(5.13)

It can be shown that for *Φ*<0, the integral converges for all *Φ* values. This regime corresponds to the expansion flows. Compression flows are characterized by the values *Φ*>0. In this case, the integral converges if 0≤*Φ*<1.

The integral may be expressed in the form(5.14)For *Φ*=0, the integral has a finite value I_{1}. Therefore, for supersonic flows for small values of the pressure disturbance, and for a small change in the boundary-layer thickness, the solution can be found in the following form:(5.15)

Boundary conditions imposed at large distances upstream give C_{2}=0. The constant *C*_{1} is not known beforehand and should be obtained taking into account an additional boundary condition imposed somewhere downstream. The value of this constant corresponds to a change in the origin of the coordinate system. The sign of the constant determines what sort of flow we are investigating—a positive sign corresponds to the compression flows and negative sign corresponds to the expansion flows.

Results of computations are presented in figures 3 and 4 where induced pressure distribution as a function of the parameter *Φ* is depicted. Negative *Φ* values correspond to expansion flows, while positive *Φ* values correspond to compression flows.

It is worth mentioning that the model of blow-off should be modified if the change in the boundary-layer thickness owing to induced pressure is comparable with the corresponding change owing to injection.

## Acknowledgments

The work was carried out under the financial support of the Russian Foundation of Basic Research (grant 02-01-00598).

## Footnotes

One contribution of 19 to a Theme ‘New developments and applications in rapid fluid flows’.

- © 2005 The Royal Society