**Shock-Induced Turbulent Boundary Layer Separation in Over-Expanded Rocket Nozzles: Physics, Models, Random Side Loads, and the Diffusive Character of Stochastic Rocket Ascent**

R. G. Keanini, T. D. Nortey, Karen Thorsett-Hill, N. Srivastava, Sam Hellman, P. T. Tkacik and P. Douglas Knight *Department of Mechanical Engineering & Engineering Science The University of North Carolina at Charlotte USA*

#### **1. Introduction**

Contrary to popular belief, and notwithstanding two hundred years of scientific study (Gruntman , 2004), the problem of accurately predicting rocket ascent remains largely unsolved. The difficulties trace to a variety of altitude-, speed-, and launch-site-dependent random forces that act during rocket ascent, including: i) aerodynamic forces (Sutton & Biblarz, 2001), ii) forces due to wind and atmospheric turbulence (Flemming et al., 1988; Justus & Johnson, 1999; Justus et al., 1990; Leahy, 2006), iii) forces produced by rocket construction imperfections (Schmucker, 1984), and iv) impacts with air-borne animals and debris (McNaughtan, 1964). Significantly, our physical understanding and ability to model the dynamical effects of each of these random features is fairly well-developed.

By contrast, understanding of the *physical origins,* as well as the *dynamical effects* of altitude-dependent, in-nozzle *random side loads,* has only recently begun to emerge (Keanini et al., 2011; Ostlund, 2002; Srivastava et al., 2010). Referring to figures 1 through 3, we find that side loads represent the end result of a chain of in-nozzle fluid dynamic processes. During low altitude flight, under over-expanded flight conditions, a pressure gradient can exist between the high pressure ambient air surrounding the rocket and nozzle, and the low pressures extant within the nozzle. This pressure gradient can force ambient air *upstream* along the nozzle wall; eventually, inertia of the ambient inflow is overcome by the pressure and inertial forces associated with the outflow, producing a near-wall recirculation region. To the supersonic flow outside the near-wall boundary layer, the recirculation zone functions as a virtual compression corner, producing an oblique shock (Keanini & Brown, 2007; Ostlund, 2002; Summerfield et al., 1954). See figures 1 through 3. Due to the altitude dependence of *Pa* = *Pa*(*H*(*t*)), where *Pa* is the ambient pressure and *H*(*t*) is the rocket's time-dependent altitude, the nominal location of the oblique shock, *xshock* = *xshock*(*H*(*t*)), also varies with altitude.

Random side loads arise due to two coupled flow features: i) The oblique shock produces a sharp, adverse pressure rise within the near-wall outflow boundary layer, forcing the boundary layer to separate from the nozzle wall; see figure 1. ii) The *shape* of the boundary

Rocket Ascent 3

<sup>157</sup> Shock-Induced Turbulent Boundary Layer Separation in Over-Expanded Rocket Nozzles: Physics, Models, Random Side Loads, and the Diffusive Character of Stochastic Rocket Ascent

**R**

φ

**s(**φ,

Fig. 3. Schematic of stochastic boundary layer separation line and associated, rippled, azimuthal oblique shock. The mean separation line position relative to the nozzle throat, *xs*, varies with rocket altitude, *H*(*t*); the corresponding nozzle radius is *R* = *R*(*H*(*t*)). The instantaneous separation line position relative to *xs*(*t*) is shown as *s*(*φ*, *t*). The separation line lies on the nozzle wall and, in a nominally symmetric nozzle, the shock forms an azimuthally independent, average angle which varies with *xs*(*t*). Adapted from (Keanini et al., 2011).

(Keanini et al., 2011; Srivastava et al., 2010) - where fore and aft pressures, determined by the shock, differ significantly - a net, time- dependent side force, or *side load*, **Fs**, is produced.

From a mass transfer perspective, a deep and unanticipated connection exists between the stochastic ascent of rockets subjected to side loading and damped diffusion processes. In order to understand the complex physical origins of this connection, it is necessary to first consider the purely mechanical features that connect shock-induced boundary layer separation to stochastic rocket response. Thus, much of this Chapter describes recent work focused on understanding these connections (Keanini et al., 2011; Srivastava et al., 2010). From a technological standpoint, the importance of separation-induced side loads derives from their sometimes catastrophic effect on rocket ascent. Side loads have been implicated, for example, in the in-flight break-up of rockets (Sekita et al., 2001), and in the failure of various rocket

I) Two stochastic models (Keanini et al., 2011; Srivastava et al., 2010) and two simple (deterministic) scaling models (Keanini & Brown, 2007) have recently been proposed to describe shock-induced boundary layer separation within over-expanded rocket nozzles (Keanini & Brown, 2007; Keanini et al., 2011; Srivastava et al., 2010). Earlier work, carried out in the 1950's and 60's and focused on time-averaged separation behavior, lead to development of the Free Interaction model of boundary layer separation (Carriere et al., 1968; Chapman et al., 1958; Erdos & Pallone, 1962; Keanini & Brown, 2007; Ostlund, 2002). Our first objective centers on describing the physical bases underlying these models, as

well as highlighting experimental evidence that supports the validity of each.

**t)**

,

**Oblique**

**Shock**

**Separation**

**Line**

**<sup>x</sup>s(H(t))**

**s L**

**1.1 Connection to mass transfer**

**1.2 Chapter objectives**

engine components (Keanini & Brown, 2007).

The objectives of this Chapter are as follows:

Fig. 1. Schematic of shock-induced boundary layer separation in rocket nozzles. The pressure variation shown is characteristic of free interaction separation problems. Adapted from Ostlund (2002).

Fig. 2. Shock-induced boundary layer separation in overexpanded supersonic nozzle flow. The process typically occurs during low altitude flight when ambient pressure is high enough to force atmospheric air into the nozzle. The incoming air flows upstream along the low-inertia, near-wall region until downstream-directed boundary layer inertia turns it, forming a virtual compression corner. An oblique shock thus forms, and the combined action of shock-induced pressure rise and inertial pressurization produced by the inflow forces the down-flow boundary layer to separate. Pressures, mach numbers, and velocities are denoted, respectively, by *P*, *M*, and *U* and *V*. Axial positions where the boundary layer starts to thicken (*i* denotes *incipient*), and where it separates are denoted, respectively, as *xi* and *xs*; the nominal shock-boundary layer interaction zone is shown as *Ls*. Since the separation line position, *xs*, and downstream conditions vary with the altitude-dependent ambient pressure, *Pa* = *Pa*(*H*(*t*)) (Keanini & Brown, 2007), all variables shown likewise vary with *H*(*t*).

layer separation line, which at any instant, forms a closed curve along the nozzle periphery, varies randomly in space and time; see figure 3. Due to relatively uniform pressure distributions extant on the up- and downstream sides of the instantaneous separation line

Fig. 3. Schematic of stochastic boundary layer separation line and associated, rippled, azimuthal oblique shock. The mean separation line position relative to the nozzle throat, *xs*, varies with rocket altitude, *H*(*t*); the corresponding nozzle radius is *R* = *R*(*H*(*t*)). The instantaneous separation line position relative to *xs*(*t*) is shown as *s*(*φ*, *t*). The separation line lies on the nozzle wall and, in a nominally symmetric nozzle, the shock forms an azimuthally independent, average angle which varies with *xs*(*t*). Adapted from (Keanini et al., 2011).

(Keanini et al., 2011; Srivastava et al., 2010) - where fore and aft pressures, determined by the shock, differ significantly - a net, time- dependent side force, or *side load*, **Fs**, is produced.
