Wall Shear Stress at Flat Surface Portions
The wall shear stress tw is the cause of the skinfriction drag, which is exerted by the flow on the flight vehicle. For a CAV this drag together with the form drag can be up to approximately 50 per cent of the total drag. For RV’s at large angle of attack, the skinfriction drag is almost negligible [36]. The wallshear stress can also considerably influence lift forces, especially during flight at higher angles of attack, and moments around the pitch and the yaw axis. Finally it is a deciding factor regarding erosion phenomena of surface coatings of thermal protection systems of RV’s.
The wall shear stress in the continuum regime and in Cartesian coordinates is defined as, see also eq. (4.36)




























We introduce p*T* = pcTc together with the powerlaw formulation of viscosity. With the nomenclature used for the thicknesses of compressible boundary layers, where the subscript ‘c’ stands for the compressible and ‘ic’ for the incompressible case, we obtain
with C = 0.332 and n = 0.5 for laminar flow, and C = 0.0296 and n = 0.2 for turbulent flow.
An alternative formulation is:
Before we summarize these results, we have a look at the result, which exact theory yields with the use of the LeesDorodnitsyn transformation for the wall shear stress of a selfsimilar compressible laminar boundary layer
[35].
We quote the result in a form given by J. D. Anderson Jr. [37]
The dependence of tw on the Reynolds number is like in eq. (7.144). The function f ‘(0) is the derivative of the velocity function f1 = u/ue at the wall. It implicitly is a function of the boundarylayer edge Mach number Me, the Prandtl number Pr, and the ratio of specific heats 7. Hence we have a dependence like in eq. (7.136) on these parameters, however implicitly and in different form.
Summary. Like for the boundarylayer thicknesses discrepancies can be found in the literature regarding the simple relations for the wall shear stress of twodimensional incompressible and compressible flatplate turbulent boundary layers. We do not pursue this problem further.
We summarize the results, eqs. (7.144), and (7.145), respectively eq. (7.146), in Table 7.5. We substitute also ‘TO’ conditions by general ‘ref’ conditions and choose ш = ш^ = 0.65 in the viscosity law, Section 4.2.[106] We introduce the dynamic pressure qref = 0.5 prefv2ef and break up all Reynolds numbers Re into the unit Reynolds number Reu and the running length x in order to show explicitly the dependencies on these parameters.
Table 7.5. Flat surface portions: dependence of boundarylayer wall shear stress on the running length x, the dynamic pressure qref, the unit Reynolds number Re’Uef and the referencetemperature ratio T*/Tref (ш = = 0.65).

The third, fourth and the fifth column in Table 7.5 give the basic dependencies of the wall shear stress of both incompressible and compressible boundary layers on the running length x, the unit Reynolds number ReUef and the dynamic pressure qref, the sixth column the dependence of the wall shear stress of compressible boundary layers on the referencetemperature ratio T*/Tref, i. e., on the wall temperature Tw and the Mach number Mref, eq. (7.129).
These dependencies give us insight into the basic behavior of the wall shear stress of twodimensional flatplate boundary layers.
— Dependence on the boundarylayer running length x.
The wall shear stress decreases with increasing x. The wall shear stress of laminar boundarylayers reduces stronger (ж x0 5) with x than that of turbulent boundary layers (ж x0 2). Remember that the thickness of the viscous sublayer grows only very weakly with x compared to the thickness of the laminar boundary layer.
— Dependence on the dynamic pressure qref.
The wall shear stress of laminar as well as of turbulent boundary layers increases linearly with increasing dynamic pressure.
— Dependence on the unit Reynolds number ReUef.
The wall shear stress depends on the inverse of some power of the unit Reynolds number. The larger ReUef, the smaller is the wall shear stress. The wall shear stress of laminar boundary layers reacts stronger on changes of the unit Reynolds number (ж (Re^^ )0’5) than that of turbulent boundary layers (ж (Re^ef )0 2). This result is not surprising, because the dynamic pressure qо was isolated, see, e. g., eq. (7.146). In the alternate formulation of eq. (7.147) we have ж (ReUef )0 5 for the laminar, and ж (ReUef )0 8 for the turbulent case.
— Dependence on T*/Tref.
Summary. In Table 7.6 the general dependencies of the wall shear stress tw in eq. (7.149) are summarized for laminar and turbulent flow. Again we choose ш = = 0.65 in the viscosity law, Section 4.2. We introduce, like
before, the dynamic pressure qref and break up the Reynolds number Re into the unit Reynolds number Reu and the radius R. Regarding the reference temperature dependencies the differentiation recommended in [16] for the laminar and the turbulent case (see above) is not made. Instead we write on the basis of eq. (7.150)
The results are:
— Dependence on the dynamic pressure q, x.
Table 7.6. Attachment line at the infinite swept circular cylinder: dependence of wall shear stress Tw, eq. (7.149), of the compressible laminar and turbulent boundary layer on the dynamic pressure qTO, the radius R, the sweep angle p, the unit Reynolds number Re^, and the referencetemperature ratio T* /Tx (ш = = 0.65).

— Dependence on the radius R.
The wall shear stress decreases with increasing R stronger for laminar, ж R0’5, than for turbulent flow, ж R0’21.
— Dependence on the sweep angle p.
For p = 0° we have the case of the nonswept circular cylinder (2D case), where Tw, scy is zero in the stagnation point. For p ^ 90° also Tw, scy ^ 0. This means that the attachment line ceases to exist for p ^ 90° and we get the situation on an infinitely long cylinder aligned with the freestream direction, where finally rw, scy becomes zero. In between we observe that Tw, scy first increases with increasing p and that stronger for laminar than for turbulent flow. This reflects the behavior of the component we of the external inviscid flow, Fig. 6.37 b), which grows stronger with increasing p than the velocity gradient due/dx across the attachment line declines with it. At large p finally the effect reverses, first for laminar then for turbulent flow, and Tw, scy drops to zero.
— Dependence on the unit Reynolds number ReЦ,.
The wall shear stress depends on the inverse of some power of the unit Reynolds number in the same way as on flat surface portions with ж (ReU)0 5 for the laminar and ж (Re^)0 21 for the turbulent boundary case. The larger ReU, the smaller is the wall shear stress. Again one has to keep in mind the dependence on the dynamic pressure, which has been isolated in this consideration.
— Dependence on T*/T, x.
At a general threedimensional attachment line the velocity gradient due/dxx=0 and are not connected explicitly to a geometrical property of the configuration. In any case it can be stated that the larger due/dx, the smaller the relevant boundarylayer thickness, and the larger the wall shear stress. Of course also here the magnitude of we plays a role. tw depends also here inversely on the wall temperature and that, like in general, stronger for turbulent than for laminar flow.
Leave a reply