Three-Dimensional Hypersonic Viscous Flows

Tollmien–Schlichting wave cancellation via localised heating elements in boundary layers

“…We first make use of the combined unsteady Dorodnitsyn–Howarth transform followed by the Prandtl transposition given by and The form of the transform given previously with the lower limit non-zero is an extension of the usual unsteady Dorodnitsyn–Howarth transform, which is discussed in van Dyke (1952) for the one-dimensional case and Neiland et al. (2007) for the two-dimensional case. The equations (2.8) and boundary conditions (2.9) reduce (see appendix A for details) to The boundary conditions are where The term involving the integral in the expression for in (2.14 f ) represents the additional displacement effect produced by the wall heating.…”

“…One of the referees kindly directed us to the work of van Dyke (1952) and the relevant page in Neiland et al. (2007) that helped us in improving the appendix when discussing the combined Prandtl–Dorodnitsyn–Howarth transformation.…”

“…Combined Prandtl–Dorodnitsyn–Howarth transform Following van Dyke (1952) and Neiland et al. (2007), we introduce the change of variables where and Note that we have introduced the independent variables for extra clarity in the following discussion, although in the main text we have retained and only. The combined Prandtl–Dorodnitsyn–Howarth transformation given by (A 1 a – c ) and (A 2) and as used in the text, is a generalisation of the more familiar Dorodnitsyn–Howarth transform, see van Dyke (1952) and Neiland et al. (2007), in which the lower limit in the integral for is zero.…”

“…(2007), we introduce the change of variables where and Note that we have introduced the independent variables for extra clarity in the following discussion, although in the main text we have retained and only. The combined Prandtl–Dorodnitsyn–Howarth transformation given by (A 1 a – c ) and (A 2) and as used in the text, is a generalisation of the more familiar Dorodnitsyn–Howarth transform, see van Dyke (1952) and Neiland et al. (2007), in which the lower limit in the integral for is zero. Details of how this transform reduces the continuity equation to one without , and how the boundary conditions at the wall are satisfied, are given in the following. Using the transformation we have Next integrating the continuity equation (2.8 a ) and making use of the boundary conditions we obtain Next we make use of the Leibniz's rule for differentiation under the integral sign in (A 5) together with the boundary conditions in (A 4) to give We now introduce and making use of the definition of , from (A 6) we have Combining (A 2) with the last result we find that This now shows that the transformed continuity equation is satisfied.…”

On quasi-steady boundary-layer separation in supersonic flow. Part 2. Downstream moving separation point

“…Later, many researchers were involved in the development of the theory, and it became clear that the viscous–inviscid interaction plays a key role in a wide variety of fluid-dynamic phenomena. An exposition of applications of the theory to different forms of the boundary-layer separation may be found, for example, in Sychev et al (1998), Neiland et al (2007) and Ruban (2018).…”