Remarks on Similarity Parameters

In the preceding sub-sections we have studied the governing equations of fluid flow. We considered in particular wall boundary conditions and similarity pa­rameters. The latter we derived in an intuitive way by comparing flow entities of the same kind, e. g., convective and molecular transport of momentum in order to define the Reynolds number.

The П or Pi theorem, see, e. g., [23], permits to perform dimensional analysis in a rigorous way. For us it is of interest that it yields parameters additional to the basic similarity parameters, which we derived above.

For the problems of viscous aerothermodynamics the ratio of wall tem­perature to free-stream temperature

T

T w

TO

is a similarity parameter [24].

A more general form is given in [19]:

Tw — Tref

Tref

This usually ignored similarity parameter is of importance, if thermal surface effects are present in the flow under consideration, Section 1.4.

Other similarity parameters are the Damkohler numbers, Section 5.4, con­cerning reacting fluid flow in general, but also the binary scaling parameter pL for flows in which dissociation occurs, see, e. g., [25, 26].

There are two aspects to deal with similarity parameters. The first is that they permit, as we did above, to identify, distinguish, and model math­ematically flow phenomena (“phenomena modeling”), for example, subsonic, transonic, supersonic and hypersonic flow.

The other aspect (“ground-facility simulation”) is that in aerothermody – namics, like in aerodynamics, experimental simulation in ground-simulation facilities is performed with sub-scale models of the real flight vehicle.

That this is possible in principle is first of all due to the fact that our simulation problems are Galilean invariant, Section 4.1. Secondly it is neces­sary that the relevant flight similarity parameters are fulfilled. This is a basic problem in aerodynamic and aerothermodynamic ground-facility simulation, because in general only a few of these parameters can be duplicated. A spe­cial problem is the thermal state of the surface. Nowadays ground-facility simulation models usually have cold and/or thermally uncontrolled surfaces.

For both aspects, however, it is important to use proper reference data for the determination of similarity parameters. For phenomena modeling pur­poses to a certain degree fuzzy data can be used. For ground-facility simu­lation the data should be as correct as possible, even when the resulting similarity parameters cannot be duplicated [27]. This is necessary in order to estimate kind and magnitude of simulation uncertainties and errors [28]. In design work margins are governed by these uncertainties and errors in concert with the design sensitivities [29].

4.3 Problems

Problem 4.1. A flight vehicle model in a ground-simulation facility has a length of 0.1 m. The free-stream velocity in the test section is 3,000 m/s. How long is the residence time tres, how long should the measurement time tmeas = tref be, if a Strouhal number Sr A 0.2 is demanded?

Problem 4.2. Compute /^suth, Mi, М2 for air at T = 500 K. What are the differences A^i and A^2 of Mi and p2 compared to MSuth?

Problem 4.3. Compute kHan, ki, k2 for air at T = 500 K. What are the differences Aki and Ak2 of ki and k2 compared to kHan? Compute k = kEucken with eq. (4.18), too. What is the difference between kHan and that value?

Problem 4.4. Compute cp for air at T = 500 K, and find 7 and Pr, the latter with kHan and with kEucken. Compare the two Pr with the result found from eq. (4.19).

Problem 4.5. Incompressible flow is defined by zero Mach number M. What does M = 0 mean?