# Displacement Thickness at a Highly Cooled Wall

In three-dimensional boundary-layer flow the displacement thickness Ji can become negative [3]. This typically happens if the flow diverges strongly, for instance at attachment lines. Regarding two-dimensional flow one expects intuitively that J1 will always be positive. However, if the wall is highly cooled, J1 may become negative. This effect is illustrated with the example of a rocket-nozzle boundary-layer flow [5]. The flow was computed with an high-temperature real-gas model, turbulent flow was assumed. We consider the situation at a location downstream of the nozzle throat. Table 10.3 gives the boundary-layer edge flow parameters (e) and the wall temperature Tw.

Table 10.3. Parameters of the nozzle-flow case with a highly cooled wall [5].

 Me Ret [1/m] мє [m/s] Ге [К] Pe [kg/m3] Tw [K] Boundary-layer state 2.23 8.071-106 3,119.51 2,418.24 0.1881 510 turbulent

The results in terms of the parameters stream-wise Mach number M, static temperature T, wall-tangential velocity u, density p, and the wall – tangential mass flux pu are given in Fig. 10.2. Except for the Mach number they each were made dimensionless with their boundary-layer edge value.

 Fig. 10.2. Rocket-nozzle flow downstream of the nozzle throat [5]. Dimensionless distributions of Mach number M, temperature T/Te, velocity u/ue, density p/pe, and mass-flow pu/(peue) across the boundary layer. The boundary-layer edge values are given in Table 10.3.

The wall to edge temperature ratio is Tw/Te « 0.2. Accordingly the den­sity ratio is pw /pe « 5 (the respective graph in the figure extends outside to the right). The mass flux pu in the boundary layer at maximum is al­most 12 per cent higher than that at the boundary-layer edge. The resulting displacement thickness, eq. (7.108), is negative with Ji = —0.27 m.[173]