Test Methods and Conditions
Chapter 4 Analysis
4.2 Semi-Empirical e N Method
Smith and Gamberoni (1956) and van Ingen (1956) proposed a boundary layer tran- sition prediction method, initially for the low-speed, incompressible case, based upon calculating the relative amplification of disturbances present in the boundary layer.
Further historical background on this method was reviewed by the latter author in van Ingen (2008). For an overview of the method’s subsequent application to high-speed compressible boundary layers, see Herbert (1997).
4.2.1 Overview
The eNmethod is based on the observation that laminar boundary layers are unstable and small disturbances can be amplified to the point that nonlinearity results in the breakdown of laminar flow and transition to turbulence. The critical amplitude of linear disturbances is a highly simplified view of transition and is one of many mechanisms by which transition may occur. The growth rates σ of the instabilities are obtained from solutions of the small disturbance equations obtained by linearizing the Navier-Stokes equations for small perturbations around the mean flow in the boundary layer as discussed in Schlichting and Gersten (2001). In the most simplified type of linear instability analysis, assuming parallel flow and temporal growth at a fixed frequency, all disturbances are assumed to vary with time and distance with amplitude
A∼φ(y) exp(i(αx−ωt)) (4.9) whereω = 2πf is the temporal frequency of oscillation, −Im(α) is the spatial growth rate and φ is an eigenfunction describing the shape of the perturbation at a given downstream location. The solution of the eigenvalue problem forφandαas a function of location within the boundary layer can be used to predict the growth in amplitude A(x) of disturbances in pressure, velocity, density and temperature with downstream distance in terms of the ratio A/A(x), where the disturbance propagates within the initial laminar boundary layer in thex-direction and A0 is an arbitrary initial ampli- tude of the disturbance. The critical value of the amplitude is linked to transition onset location by empirical observation with a critical value of N at the transition
location, where eNTr ≈ATr/A0. The critical value of N is determined empirically, as discussed subsequently, and is known to be a strong function of factors such as the freestream turbulence level, wall roughness, and pressure gradient. As described in Johnson (2000) Chapter 5, N represents the integrated growth rate of boundary layer disturbances and is defined as:
N(f, x) = ln A
A0
= x x0
σdx
wheref is the disturbance frequency,xrepresents distance along the surface of interest (x0 is the location of instability onset) and the disturbance growth rateσ is:
σ(f, x) =−Im(α(ω, x)) + 1 2E
dE dx
where α is the complex wave number, computed at each streamwise location, and E is the kinetic energy of the disturbance, integrated over the spatial variable in the normal direction to the flow, n:
E =
n
¯ ρ
|u|2+|v|2+|w|2 dn
The amplification rate α and the disturbance amplitudes u, v and w are com- puted either with a linear stability analysis (LST) assuming locally parallel flow in the boundary layer or else the parabolized stability equations (PSE) that approximately account for the non-parallel development of the flow; see Johnson (2000) for how this is implemented in the STABL software.
The change inσand N with distance along the boundary layer reflects the underly- ing stability characteristics. At a given downstream location boundary layers are only unstable over a narrow range of frequency or spatial wavelength (see Figure 4.6 and as- sociated discussion). The center of the band of unstable frequencies shifts downward
as the boundary layer becomes thicker with increasing distance downstream. Starting from the leading edge and progressing downstream, a disturbance is initially stable (σ <0) and damped, becomes unstable (σ > 0) when the neutral stability boundary is crossed at x0 and grows, then again becomes stable and damped. This results in amplitudesA and N factors that increase with increasing downstream distance (after reaching the neutral stability boundary), reach a maximum value, then decrease. The maximum value of N increases with increasing downstream distance due to the longer distance available for positive growth. Figure 4.3, calculated for shot 2742 from the present study, illustrates this behavior in a series of N factor curves at fixed frequency along the surface of a five-degree half-angle cone. The overall N factor curve is taken to be the envelope of these individual curves. See Wagnild (2012) Chapter 3 for a more detailed description of the implementation of the eNmethod used in the present study.
The chief weakness of the eNmethod is that it does not account for the amplitude of the initial disturbance from which N derives, but only the subsequent growth rate of disturbances. Thus, the most useful comparisons can only be made between exper- iments performed in very similar flow environments (i.e., the same geometry in the same facility), for which boundary layer receptivity characteristics may be reasonably assumed to be similar (Saric et al., 2002). The eN method also does not account for bypass transition mechanisms. However, within these limitations and with the caveat that computations become much more challenging as conditions proceed away from the two-dimensional incompressible flows for which the theory was originally devel- oped, the eN method is currently considered the most reliable technique available for estimating transition location given a repeatable freestream turbulence level and a streamlined body with a relatively smooth surface. In particular, it provides a means of relating boundary layer characteristics across different geometries, thermochemical models, and freestream conditions. For “noisy” tunnels, as some authors, including
0 0.2 0.4 0.6 0.8 1 0
2 4 6 8 10 12 14 16 18
x [m]
N Factor
1493 kHz 1758 kHz
2110 kHz
1229 kHz
Figure 4.3: Computed N factor at selected frequencies along the surface of a 5◦ half- angle cone, from shot 2742. Four individual N factor results, in red, are labeled with their frequencies to illustrate the frequencies contributing to the maximum N factor at different positions on the cone.
Schneider (2001), have characterized Caltech’s T5 facility, the N factors at transition are around 4 to 7 (Johnson, 2000). For so-called “quiet” tunnels, the characteristic transition N factors may be 8 to 9, extending all the way up to N factors equivalent to free flight of 10 to 11 (Schneider, 2008a). For N∼20 and above, the effects of Brow- nian motion alone are enough to cause transition (Fedorov, 2011b). The critical value of N is empirical and depends, among other factors, on the disturbance environment;
therefore, NTr must be calibrated for a particular wind tunnel facility.
4.2.2 Past Work
Historically, many computational studies on hypersonic boundary layer instability with the eN method have been based upon linear stability theory (LST), which ne- glects the nonparallel nature of a the boundary layer as well as non-linear effects.
Malik and Anderson (1991), for example, used this approach to demonstrate that first-mode instabilities were suppressed, and second-mode instabilities amplified, for real-gas hypersonic flows at Mach 10 and Mach 15. Johnson et al. (1998) used LST to examine the effects of freestream total enthalpy and chemical composition on tran- sition onset location for T5 flows. Reed et al. (1996) review the application of LST to boundary layers. For an overview of hypersonic boundary layer stability in the context of transition prediction methods, see Fedorov (2011a).
The eN method implemented with parabolized stability equations (PSE) has be- come popular in recent decades as a more accurate representation of boundary layer instabilities than that available from LST (Herbert, 1997). Malik (2003) used PSE methods to analyze second-mode growth and transition in two flight experiments, in- cluding Reentry-F (Zoby and Wright, 1977). More recently, Wagnild et al. (2012) and Gronvall et al. (2014) applied the eN method as implemented in STABL/PSE-Chem to a notional cone at Mach 12, and used the same software to analyze transition data from cones at Mach 6.7–7.7 in the Japan Aerospace Exploration Agency’s free-piston High Enthalpy Shock Tunnel, respectively. As part of the present study, Wagnild et al. (2010) used the eN method to evaluate the effect of gas injection into a conical boundary layer on transition onset location, and Jewell et al. (2013c) examined the effect of freestream gas mixtures with the same geometry.