# Attached Viscous Flow

7.1.1 Attached Viscous Flow as Flow Phenomenon

The flow past a body exhibits, beginning at the forward stagnation point, a thin layer close to the body surface, where viscous effects play a role. They are due to the fact that the fluid in the continuum regime fully sticks to the surface: no-slip boundary condition, eq. (4.47), or—in the slip-flow regime— only partly: slip-flow boundary condition, eq. (4.48). In these cases we speak about attached viscous flow. Away from this layer the flow field is inviscid, i. e., viscous effects can be neglected there. Of course the inviscid flow field behind the flight vehicle, and at large angle of attack also above it, contains vortex sheets and vortices, which are viscous phenomena, together with shock waves, Section 6.1.

Attached viscous flow must always be seen in connection with the flow past the body as a whole. The body surface with either no-slip or slip boundary condition, together with the thermal and thermo-chemical boundary conditions, Section 4.3, is the causa prima for that flow. If the layer of attached viscous flow is sufficiently thin, the flow in it is nearly parallel to the body surface and the gradient of the pressure in it in direction normal to the surface vanishes. Attached viscous flow in this case is governed by the pressure field of the external inviscid flow field, and, of course, its other properties, and the surface boundary conditions. We call such an attached viscous flow sheet a “boundary layer”.[84] The boundary layer can be considered as the phenomenological model of attached viscous flow [1]. In the following we use the terms “attached viscous flow” and “boundary layer” synonymously.

Of special interest in flight vehicle design are the (flow) boundary-layer thicknesses S, the thickness of the viscous sub-layer Svs, the displacement thickness S1, the wall shear stress tw, and the thermal state of the surface, which encompasses both the wall temperature Tw and the heat flux in the gas at the wall qgw. All for laminar and for turbulent flow. They will be discussed in detail in the following sections.

For these discussions we assume in general two-dimensional boundary – layer flows. In reality the flows past hypersonic flight vehicles are threedimensional, see, e.g., Figs. 7.8, 7.9, and 9.5. However, large portions of the flow can be considered as quasi two-dimensional, and can be treated, with due caution, with the help of two-dimensional boundary-layer models, for instance for initial qualitative considerations, and for the approximate determination of boundary-layer parameters. This does not hold in regions with attachment and separation phenomena etc. [1], which, for example, are present in flight at high angle of attack, Fig. 9.5.

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