Supersonic Engine Inlet

Supersonic Engine Inlet

Background
Benchmark ID	supersonicEngineInlet_n11730m2q1
Category	oberwolfach
System-Class	LTI-FOS
Parameters
nstates	11730
ninputs	2
noutputs	1
nparameters	0
components	A, B, C, E
Copyright
License	NA
Creator	Steffen Werner
Editor	Steffen Werner Christian Himpe
Location
NA

Description: Active Control of a Supersonic Engine Inlet

Figure 1: Steady-state Mach contours inside diffuser. Freestream Mach number is 2.2.

This example considers unsteady flow through a supersonic diffuser as shown in Fig. 1. The diffuser operates at a nominal Mach number of $2.2$, however it is subject to perturbations in the incoming flow, which may be due (for example) to atmospheric variations. In nominal operation, there is a strong shock downstream of the diffuser throat, as can be seen from the Mach contours plotted in Figure Fig. 1. Incoming disturbances can cause the shock to move forward towards the throat. When the shock sits at the throat, the inlet is unstable, since any disturbance that moves the shock slightly upstream will cause it to move forward rapidly, leading to unstart of the inlet. This is extremely undesirable, since unstart results in a large loss of thrust. In order to prevent unstart from occurring, one option is to actively control the position of the shock. This control may be effected through flow bleeding upstream of the diffuser throat.

A complete description of the benchmark and some model reduction results can be downloaded as PDF file here.

Active Flow Control Setup

Figure 2: Supersonic diffuser active flow control problem setup.

Fig. 2 presents the schematic of the actuation mechanism. Incoming flow with possible disturbances enters the inlet and is sensed using pressure sensors. The controller then adjusts the bleed upstream of the throat in order to control the position of the shock and to prevent it from moving upstream. In simulations, it is difficult to automatically determine the shock location. The average Mach number at the diffuser throat provides an appropriate surrogate that can be easily computed. There are several transfer functions of interest in this problem. The shock position will be controlled by monitoring the average Mach number at the diffuser throat. The reduced-order model must capture the dynamics of this output in response to two inputs: the incoming flow disturbance and the bleed actuation. In addition, total pressure measurements at the diffuser wall are used for sensing.

CFD Formulation

The unsteady, two-dimensional flow of an inviscid, compressible fluid is governed by the Euler equations. The usual statements of mass, momentum, and energy can be written in integral form as

$\begin{align} \frac{\partial}{\partial t}\iint\rho\mathrm{d}V + \oint\rho Q\cdot\mathrm{dA} & = 0,\\ \frac{\partial}{\partial t}\iint\rho Q\mathrm{d}V + \oint\rho Q (Q\cdot\mathrm{dA}) + \oint p \mathrm{dA} & = 0,\\ \frac{\partial}{\partial t}\iint\rho E\mathrm{d}V + \oint\rho H (Q\cdot\mathrm{dA}) + \oint p Q\cdot\mathrm{dA} & = 0, \end{align}$

where $\rho$, $Q$, $H$, $E$, and $p$ denote density, flow velocity, total enthalpy, energy, and pressure, respectively. The CFD formulation for this problem uses a finite volume method and is described fully in and . The unknown flow quantities used are the density, streamwise velocity component, normal velocity component, and enthalpy at each point in the computational grid. Note that the local flow velocity components $q$ and $q^{\perp}$ are defined using a streamline computational grid that is computed for the steady-state solution. $q$ is the projection of the flow velocity on the meanline direction of the grid cell, and $q^{\perp}$ is the normal-to-meanline component. To simplify the implementation of the integral energy equation, total enthalpy is also used in place of energy. The vector of unknowns at each node $i$ is therefore

$x_{i} = \begin{bmatrix} \rho_{i}, & q_{i}, & q^{\perp}_{i}, & H_{i} \end{bmatrix}^{T}.$

Two physically different kinds of boundary conditions exist: inflow/outflow conditions, and conditions applied at a solid wall. At a solid wall, the usual no-slip condition of zero normal flow velocity is easily applied as $q^{\perp} = 0$. In addition, we will allow for mass addition or removal (bleed) at various positions along the wall. The bleed condition is also easily specified. We set

$q^{\perp} = \frac{\dot{m}}{\rho},$

where $\dot{m}$ is the specified mass flux per unit length along the bleed slot. At inflow boundaries, Riemann boundary conditions are used. For the diffuser problem considered here, all inflow boundaries are supersonic, and hence we impose inlet vorticity, entropy and Riemann’s invariants. At the exit of the duct, we impose outlet pressure.

Linearized CFD Matrices

The two-dimensional integral Euler equations are linearized about the steady-state solution to obtain an unsteady system of the form

$\begin{align} E\dot{x}(t) & = Ax(t) + Bu(t),\\ y(t) & = Cx(t). \end{align}$

The descriptor matrix $E$ arises from the particular CFD formulation. In addition, the matrix $E$ contains some zero rows that are due to implementation of boundary conditions.

Origin

This benchmark is part of the Oberwolfach Benchmark Collection; No. 38866, see also .

Data

The matrices of this benchmark can be downloaded in the Matrix Market format:

• SupersonicEngineInlet-dim1e4-Inlet.tar.gz.

The size of the file is 5.4 MB. The matrix name is used as an extension of the matrix file.

Dimensions

System structure:

$\begin{align} E \dot{x}(t) & = Ax(t) + Bu(t),\\ y(t) &= Cx(t) \end{align}$

System dimensions:

$E \in \mathbb{R}^{11730 \times 11730}$, $A \in \mathbb{R}^{11730 \times 11730}$, $B \in \mathbb{R}^{11730 \times 2}$, $C \in \mathbb{R}^{1 \times 11730}$

Citation

To cite this benchmark, use the following references:

• For the benchmark itself and its data:

Oberwolfach Benchmark Collection, Supersonic Engine Inlet. hosted at MORwiki - Model Order Reduction Wiki, 2005. http://modelreduction.org/index.php/Supersonic_Engine_Inlet

@MISC{morwiki_supsonengine,
  author =       {{Oberwolfach Benchmark Collection}},
  title =        {Supersonic Engine Inlet},
  howpublished = {hosted at {MORwiki} -- Model Order Reduction Wiki},
  url =          {http://modelreduction.org/index.php/Supersonic_Engine_Inlet},
  year =         2005
}

• For the background on the benchmark:

@MASTERSTHESIS{morLas13,
  author =       {G. Lassaux},
  year =         2002,
  title =        {High-Fidelity Reduced-Order Aerodynamic Models: Application to
                 Active Control of Engine Inlets},
  school =       {Massachusetts Institute of Technology},
  address =      {Cambridge, USA},
  url =          {http://web.mit.edu/kwillcox/Public/Web/LassauxMS.pdf}
}