**1.1 Connection to mass transfer**

2 Will-be-set-by-IN-TECH

**shock**

**−boundary**

**layer interaction**

**zone**

**recirculation**

**zone**

**x**

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

> **V2 <sup>M</sup> U2 <sup>P</sup> <sup>2</sup> <sup>2</sup>**

**xi xs**

**Ls** 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

**recirculation zone layer separated boundary**

**Oblique Shock**

β

**Separated Boundary Layer**

θ **Zone**

**Recirculation**

**oblique shock**

**wall pressure (vacuum) wall pressure (sea−level) nozzle exit pressure ambient pressure**

**layer boundary**

**psi xxx <sup>r</sup>**

**nozzle**

**wall**

**Pi Mi**

δ **i**

**Nozzle Wall**

**U**

**i**

**nozzle radius**

Ostlund (2002).

**wall pressure**

**P**

**a P e P p P s P i P**

> 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 engine components (Keanini & Brown, 2007).

#### **1.2 Chapter objectives**

The objectives of this Chapter are as follows:

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.

Rocket Ascent 5

<sup>159</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

the ambient pressure, *Pa*. The time-average separation point, *xs*, lies immediately upstream of

Of central importance in nozzle design is determining both the conditions under which separation will occur and the approximate separation location. A number of criteria have been proposed for predicting the nominal free shock separation point, *xs*; see, e.g., (Keanini & Brown, 2007; Ostlund, 2002) for reviews. Since the boundary layer pressure rise between *xi* and *xs* depends primarily on the inviscid flow Mach number, *Mi*, most criteria relate either a gross separation pressure ratio, *Pi*/*Pa*, or more recently, a refined ratio, *Pi*/*Pp*, to *Mi* (Ostlund, 2002). Given the separation pressure ratio, the separation location can then be determined

Although the actual separation process is highly dynamic, the scaling model focuses on time average flow dynamics in the vicinity of the shock interaction zone. In order to provide physical context, we briefly review the dynamical features associated with free shock separation and note simplifying assumptions made. Shock motion over the shock interaction zone appears to be comprised of essentially two components: i) a low frequency, large scale motion produced by flow variations downstream of the separation point, and occurring over the length of the shock interaction zone, *lp* = *xp* − *xi*, at characteristic frequencies, *fs* [on the order of 300 to 2000 Hz in the case of compression ramp and backward facing step flows (Dolling & Brusniak, 1989)], and ii) a high frequency, low amplitude jitter produced by advection of vortical structures through the shock interaction zone (Dolling & Brusniak, 1989). The scaling models in (Keanini & Brown, 2007) limit attention to time scales that are long relative to *f* <sup>−</sup><sup>1</sup> *<sup>s</sup>* . In addition, the model assumes that the flow is statistically stationary and

The time average pressure gradient over the shock interaction zone (*xi* ≤ *x* ≤ *xp*), given

*<sup>∂</sup><sup>x</sup>* <sup>∼</sup> *Pp* <sup>−</sup> *Pi lp*

in reality reflects the intermittent, random motion of the shock between *xi* and *xp*; see, e.g., (Dolling & Brusniak, 1989). As the shock-compression wave system oscillates randomly above (and partially within) the boundary layer, the associated pressure jump across the system

are the characteristic boundary layer thickness and temperature in the vicinity of *xi*. Under typical experimental conditions, *<sup>τ</sup><sup>s</sup>* is much shorter than the slow time scale, *<sup>f</sup>* <sup>−</sup><sup>1</sup> *<sup>s</sup>* (where *<sup>τ</sup><sup>s</sup>* <sup>≈</sup> 1 to 10*μs*); thus, the instantaneous separation point essentially tracks the random position of the shock-compression wave system, where the position of the separation point is described by a Gaussian distribution over the length of the interaction zone (Dolling & Brusniak, 1989).

In the vicinity of the separation point,*xs*, we recognize that a fluid particle's normal acceleration component within the separating boundary layer is determined by the normal component of the pressure gradient across the separating boundary layer. Thus, balancing

where *Vs* is the particle speed in the streamwise (*s*-)direction, and *R*−<sup>1</sup> is the local streamline curvature. The curvature can be evaluated by first defining the shape of the boundary layer's

*ρ V*2 *s <sup>R</sup>* <sup>∼</sup> *<sup>∂</sup><sup>P</sup>* (1)

<sup>√</sup>*kRTi*, where *<sup>δ</sup><sup>i</sup>* and *Ti*

*<sup>∂</sup><sup>n</sup>* (2)

*∂P*

is transmitted across the boundary layer on a time scale *τ<sup>s</sup>* ∼ *δi*/

using an appropriate model of flow upstream of separation.

that the separation process is two-dimensional.

**2.2 Scale analysis of shock-induced separation**

these terms (in the Navier- Stokes equations) yields:

approximately by

*xp*.

