Transition to turbulence in two-dimensional wakes

Abstract

A new technique is advanced for predicting the location of the laminar turbiiient transition zone in twodimensional wake flows. The technique is based on postulating a minimum turbulence Reynolds number required for sustaining the turbulence once it occurs. Using existing wake turbulence data and compressible laminar wake theory, the turbulence Reynolds number is then algebraically expressed in terms of freestream or wake edge conditions and integral properties. An example is calculated using laminar wake similarity, and a simple formula is found by which the transition distance is linked to the stream Mach number, the heat transfer rate, ratio of specific heats, and other parameters. Some of the existing transition data are utilized to find that the minimum or threshold turbulence Reynolds number is about 15, whereas other available data are compared and found in good agreement with the resulting formula. NomenclatureB = a constant, see Eq. (20) C } -3. constant in the fluctuation formula, Eq. (5) C 2 = a constant in the scale formula, Eq. (6) £3 -C}C2 C D h -momentum defect of wake d = diameter of wake-shedding body k = exponent in the viscosity-temperature relation, Eq. (11) M = Mach number Q = dimensionless thermal power exchanged, Eq. (16) Q = dimensional thermal power exchanged between body and flow Re = Reynolds number Re 0 = minimum Reynolds number for transition Re w = wake Reynolds number (based on edge properties and C^/2) •Re do = defined by Eq. (9) Re A = turbulence Reynolds number Re^0 = minimum possible turbulence Reynolds number Re xT = Reynolds number based on edge properties and X T t = temperature defect, = [T(Q)-T e ]/T e T = static temperature u = velocity x = distance from the wake-shedding body x = nondimensional distance -xiC D h y = ratio of specific heats A = integral scale of turbulence v -kinematic viscosity p = density a = Prandtl number ( )' • = rms wideband fluctuation (• )"' = freestream properties ( ) e = wake edge properties ( )(0) = axis properties ( ) T = properties at the transition point (zone)

Necessary Conditions for Transition in a Free Shear Layer.

“…The theoretical approach to predicting transition in the FSL is the same as utilized earlier by this author [1,4], for deriving analogous equations for the wake and boundary layer. The central statement of this method is that the turbulence Reynolds number Thus, according to the above, the present task consists of computing the turbulence Reynolds number, given by Equation (1), along the FSL.…”

“…On the other hand, in applying the theory to wakes, [1], successful predictions were made using the ReA magnitude in the center of the flow. Therefore, despite the misgivings of the preceding paragraph, we will here take the following position: if a turbulent FSL has a maximum Re A equal to or larger than the threshold ReAo, then that flow is permitted to be locally turbulent.…”

“…It is frequently mentioned that a proper measure of the effect of viscous forces on turbulence is the turbulence Reynolds number defined by Equation (1).…”

“…This agreement justified the risk of associating the necessary condition wit. the transition phenomenon, and it was so proposed in the journal publicat-on which followed [1].…”

Experimental studies of stability and transition in high-speed wakes

“…However there have not been many experiments on the stability of a wake at supersonic flow speeds carried out so far (similar studies in a boundary layer are more numerous -see, for example, Lysenko 1993;Lysenko & Maslov 1984; and many others), though such works as Behrens (1968), Behrens & Ko (1971), Behrens, Lewis & Webb (1971), Demetriades (1964Demetriades ( , 1978Demetriades ( , 1990, McLaughlin (1971), McLaughlin et al (1971) have become classical. In them the development of the natural disturbances is studied.…”