The Use of Transition Region Characteristics to Improve the Numerical Simulation of Heat Transfer in Bypass Transitional Flows

A method is presented for improving the numerical prediction of bypass transition heat transfer on a flat plate in a high-disturbance environment with zero or favorable pressure gradient. The method utilizes low Reynolds number k-e turbulence models in combination with the characteristic parameters of the transition region. The parameters representing the characteristics of the transition region used are the intermittency, transition length and turbulent spot properties. An analysis is made of the transition length in terms of turbulent spot variables. The nondimensional spot formation rate, required for the prediction of the transition length, is shown by the analysis to be a function of the spot spreading angle, the dimensionless spot velocity ratio and the dimensionless spot area ratio. The intermittency form of the k-e equations was derived from conditionally averaged equations which have been shown to be an improvement over global-time-averaged equations for the numerical calculation of the transition region. The numerical predictions are in general good agreement with the experimental data and indicate the potential use of the method in accelerating flows. Turbulence models of the k-e type are known to underpredict the transition length. The present work demonstrates how incorporating transition region characteristics improves the ability of two-equation turbulence models to simulate bypass transition for flat plates with potential application to turbine vanes and blades.

Fullmodeling ofthistransition phenomena forthepurpose ofpredicting wallshear stress andheat transfer on turbine blades must consider theeffects offree-stream turbulence, pressure gradient, streamwise curvature, surface roughness, wall/free-stream temperature ratio and flow disturbances (e.g., wakes). The present work, a first step, concentrates on the effects of free-stream turbulence and pressure gradient on a flat plate. The objective is to demonstrate a modeling approach to the transition region that will improve the numerical prediction of that region.
Properly used, Low Reynolds Number k-e turbulence models for calculating the transition heat transfer in environments of high disturbance appear to simulate the onset of transition governed by the transport of turbulence energy from the free stream into the boundary layer. However, while two-equation turbulence models predict an acceptable value for transition onset, the resulting process is generally too abrupt, resulting in an underprediction of the transition length (e.g., Rodi and Scheuerer, 1985;and Simon, 1993). The experimental evidence (Volino and Simon, 1993) suggests that the physics of the transition region is characterized by an incomplete development of the cascade of energy from large to small scale turbulence and the k-E model does not completely account for this. Three modifications of the k-e model, as a potential means of improving the simulation of transition physics, are reviewed by Simon (1993). The three methods are (1) the use of a multi-time-scale k-e model (Crawford, 1992), (2) modification of the turbulent production term in the differential equation for the turbulent kinetic energy (Schmidt and Patankar, 1988), and (3) the use of intermittency in the k-E equations (Simon and Stephens, 1991 Simon and Stephens (1991) using the method of Narasimha (1985). It expresses the transition length in terms of a nondimensional spot formation rate (N) which is assumed to be constant for flee-stream turbulence levels greater than about 1 percent. Simon and Stephens followed the approach of Narasimha and determined the value of N based on experimental data. This permitted a calculation of transition length in terms of the momentum Reynolds number for transition onset. The transition length may also be determined using an empirical approach (e.g., Dhawan and Narasimha, 1958;and Narasimha, 1978) or by the use of turbulent spot analyses (Walker and Gostelow, 1990;Walker, 1989;McCormick, 1968).
The present work follows the line of reasoning given in the latter analytical works to develop a model for the nondimensional spot formation rate (N) for a prediction of transition length and intermittencies required for the equations of Simon and Stephens. The equation developed for predicting of the transition length is expressed in terms of turbulent spot characteristics of spreading angle, velocity, and physical dimensions. A check of the analytical formulation is made using the recent turbulent spot data of Clark, LaGraff, Magari, and Jones (1992).
Having a method to predict transition length and intermittency permits the prediction of transition heat transfer. In this work numerical predictions are made for transition heat transfer, using the modified Jones-Launder turbulence model of Simon and Stephens (1991) with the TEXSTAN boundary layer computer code (Crawford, 1985). The numerical predictions are compared with the flat plate, zero, and favorable pressure gradient data of Werle (1980, 1981 The beginning of the transition region is defined by the first appearance of turbulence spots. The transition region is characterized by the intermittent appearance of these turbulent spots, which grow as they move downstream until they finally merge to form the turbulent boundary layer. The transition region, therefore, may be described as having turbulent spots with laminar-like fluid surrounding the spots. Vancoille and Dick (1988)  obtain global values of velocity and temperature. The present work refines the previous work of Simon and Stephens (1991) by removing the need to use an empirical nondimensional spot formation rate (N) for prediction of transition length and by extending the prediction of N to include pressure gradient. A brief review of the key equations follow.
As shown by Simon and Stephens, when the modeled equations for the turbulent and laminar zones are combined and the assumption made of negligible Reynolds stresses in the "laminar" zone, the following momentum and energy equations are obtained: (1) = ayL ay- Equations (1) to (6) are the simplified equations for the transition region derived by Simon and Stephens (1991 ).
Constants for the turbulence model are given in Table I. These equations require the specification of intermittency.
Equation (1) shows the Reynolds stresses multiplied by the intermittency factor. This is the form utilized by McDonald and Fish (1973) and others.

Intermittency
Specification of intermittency requires knowledge of the transition path in terms of transition start and of the transition length. Narasimha (1957), utilizing a concentrated breakdown hypothesis, derived the following transition path equation from the turbulent spot theory of Emmons (1951): Equation (7) has been compared to experimental data and has been found to be valid. Figure 1, from Volino and Simon (1991), compares the experimental data from Kim (1991), Kuan and Wang (1990), Sohn, O'Brien, and Reshotko (1989), and Blair and Anderson (1987) with Eq. (7). Figure 1 is an example of the good agreement that has been found with experimental data and Eq. (7).
Using the approach of Narasimha (1985) the transition length may be expressed in terms of the transition Reynolds number and a nondimensional spot formation rate as follows: Where N, the nondimensional spot formation rate is defined as In what follows an analysis is made of spot inception and growth. Spot inception is described in terms of the onset position and frequency of disturbance. The frequency of disturbance is taken to be the Tollmien-Schlichting wave frequency having the maximum amplification. This permits an expression for the dimensionless spot formation rate (N) in terms of the spot spreading angle, spot velocity, and spot physical dimensions. The transition length may then be calculated using Eq. (8).

Spot Formation Rate (N) Analysis
Chen and Thyson (1971 ) used the observations obtained from stability experiments with controlled disturbances, that prior to laminar flow breakdown the wave motions become three-dimensional with regularly spaced peaks and valleys. Breakdown or the appearance of turbulent spots is assumed to occur at the peaks. The turbulent spots grow and spread and eventually meet to form a fully developed (albeit immature) turbulent boundary layer. This process is illustrated in Fig. 2. The transition length, defined as the length from inception to the first meeting of adjacent spots, was shown by McCormick (1968) to be related to the distance between turbulent spots at onset (see Fig. 2) as follows: From the following intermittency-path equation derived by Narasimha (1957) We obtain for = 0.99 at x = x E Based on the assumption that breakdown to a turbulent spot occurs at the peak for each cycle of wave motion, Chen and Thyson (197 i) state that the spot formation rate (n) can be related to the frequency of disturbance (f) and the spanwise wavelength. This is stated here in terms of the distance between onset of spots (d) as follows: So that by combining Eqs. (13) and (14) we have f2°" n -4tan 2 _x4'61Ue The dimensionless spot formation rate as defined by Narasimha (1985) is given by Eq. (9), which can be reformulated and expressed as n°'v2 Re3t_ N = --7;3---Ue Theequation forthelocus ofmaximum amplification rates overthewhole range oftheFalkner-Skan pressure gradient parameter (including zeropressure gradient) asgiven byWalker is asfollows: Walker presents experimental evidence, for a free-stream turbulence of 0.3 percent, to justify employment of Eq.
(l 8). Roach and Brierley (1992) Equation (21) for N is only a function of turbulent spot characteristics and appears to justify the use of a constant value for N as suggested by Narasimha (1985). Narasimha (1985) found that for freestream turbulence levels greater than 0.1 percent the value of N has the approximate constant value of 0.7x 10 -3. However, using an 11°spot spreading angle as determined by Schubauer and Klebanoff (1955), a value of A = 2 as estimated by Emmons (1951) ) and (21). As shown in Fig. 3 the calculated N values cover the range of the experimental data, with a preference for the lower N value of 0.29x 10 3. In the above calculations for N at zero pressure gradient, a spreading angle of 11°was assumed for the low Mach number conditions of the experiments. Clark, Jones, and LaGraff (1994) report that the spot spreading angle decreases monotonically with an increase in the Mach number. Their experimental data is shown in Fig. 4. Extrapolation of their data to the low Mach number conditions of the experiments of Fig. 3 indicates that the use of an 11°spreading angle is justified.    (22) and (23) may be graphically represented in the manner suggested by Dhawan and Narasimha (1958). This is done in Fig. 5 for a number of experimental data sets given in a survey report by Volino and Simon (1991). value of k = 0.63x10 -6 was calculated at onset for the Blair and Anderson (1987) experiment (k = 0.75x10-6). This helps to justify the use of a 5°spreading angle for obtaining Eq. (23). Figure  5 demonstrates that with an increase in flow acceleration there is an apparent increase in transition length and that this increase in transition length is consistent with the characteristics of turbulent spots under accelerating conditions. The above is not considered to be a definitive test for the use of Eq. (21), but points to the need for a further study of turbulent spots subject to conditions of pressure gradient.

Heat Transfer
Calculations A calculation of the transition region was made utilizing Eqs.
(1) to (7) and Eqs. (22) and (23) for estimating the transition length. These equations were numerically solved by using the TEXSTAN boundary layer computer code (Crawford, 1985). TEXSTAN is based on the STAN5 boundary layer program developed by Crawford and Kays (1976 The zero boundary condition for the dissipation was made possible by Jones and Launder (1973) by including additional terms in the turbulence energy (Eq. (5)).
Freestream boundary_ conditions. The freestream conditions may be expressed as a set of coupled ordinary differential equations as follows: Ue dke = -e e (27) 2 de e e (28) Ue _XX--C2 k-_ Where initial values of the free-stream turbulence kinetic energy are determined from the turbulence intensity as follows: k e = 1.5(Tu eue )2 (29) jl/3(u '2 +v '2 +w '2) where Tu e = (30) Ue And the initial dissipation rate determined from the following: The dissipation length scale (Le) can be calculated from the longitudinal integral length scale using the method of Hancock (1980). Initial values. To initiate calculations, a Blasius velocity profile and a flat temperature profile (unheated start) were utilized. For the turbulence transport equations the following initial profiles suggested by Rodi and Scheuerer (1985) were used: And e=0.35k--

RESULTS AND DISCUSSION
Calculations are presented for turbulence levels of 1.4 and 2.8 percent at zero pressure gradient and 2.2 percent for a favorable pressure gradient with an acceleration parameter (k) of 0.75x10 -6. It was necessary, to achieve good numerical results, to use about 100 grid points in the cross-stream directions. Comparisons of the calculations were made with the experimental data of Werle (1980, 1981). These experiments were conducted in a low speed wind tunnel (30 ft/sec to 130 ft/sec) under ambient air conditions utilizing grids for turbulence generation. A favorable pressure gradient was produced by the use of wedge in the test section. The test section was heated at a rate of 0.078 Btu/ft2-sec with an unheated length of 0.141 ft. The experimental boundary conditions used in the calculations are listed in Table IV Fig. 6(b). There is generally good agreement between experiment and prediction. Note in Fig. 7   transfer. This is also true in the case with pressure gradient (Fig. 8). Figure 8 shows that, for the case of pressure gradient, there is general agreement between experiment and prediction, although the computed curve appears abrupt in the region of the minimum heat transfer. There is clearly a need to improve the model.

CONCLUDING REMARKS
A demonstration has been made of the important role played by incorporating the characteristics of the transition region to provide improved modeling for predictive codes. A model has been developed for predicting the nondimensional spot formation rate (N) in terms of turbulent spot variables. This model allows for a determination of the transition length and intermittency in terms of the turbulent spot velocity, turbulent spot spreading angle, and the turbulent spot physical dimensions. The model produced values of N which are in acceptable agreement with the values obtained from zero pressure gradient experiments. There is a need to further confirm the present approach by investigating the behavior of turbulent spots under conditions of pressure gradient and curvature for a determination of their relationship to transition length and intermittency. It is expected that the above information will aid the transition modeling efforts for the improved prediction of shear stress and heat transfer on turbine blades.