This paper considers the numerical optimization of a double ramp scramjet inlet using magnetohydrodynamic (MHD) effects together with inlet ramp angle changes. The parameter being optimized is the mass capture at the throat of the inlet, such that spillage effects for less than design Mach numbers are reduced. The control parameters for the optimization include the MHD effects in conjunction with ramp angle changes. To enhance the MHD effects different ionization scenarios depending upon the alignment of the magnetic field are considered. The flow solution is based on the Advection Upstream Splitting Method (AUSM) that accounts for the MHD source terms as well. A numerical Broyden-Flecher-Goldfarb-Shanno- (BFGS-) based procedure is utilized to optimize the inlet mass capture. Numerical validation results compared to published results in the literature as well as the outcome of the optimization procedure are summarized to illustrate the efficacy of the approach.

Scramjet engine inlet flow is subject to many engineering tribulations, mainly due to the fact that the base geometry of the engine is suited to a very narrow range of flight conditions. In order to have the engine operate efficiently across a broader range of flight conditions the inlet flow must be adjusted. One obvious method to tune the inlet flow is a mechanically actuated, variable geometry inlet that can adjust its shape in flight to achieve optimal inlet conditions [

The scramjet engine is designed for hypersonic flight with supersonic combustion and is flown at speeds ranging approximately from Mach

It is to be mentioned that initial studies of optimizing duct flows for MHD power generators [

Previous studies on the effects of MHD control on inlet flow optimization [

Symmetric double ramp full inlet.

The main purpose of this study is to increase the mass capture of a double ramp cowl style scramjet inlet. The performance metric is optimized using a combination of changing geometry (ramp angles) and including MHD effects. The discretized Euler's equations are augmented with MHD source terms and the flow solution is obtained via the AUSM method. The flow solution methodology is validated against test cases drawn from literature available in the public domain. The optimization methodology is also validated together with the flow solver on the traditional double ramp inlet problem with and without MHD terms. Finally, the optimization results for the combined MHD

The double ramp scramjet inlet studied in this paper is a cowl style inlet. Such an inlet represents a forebody external compression region followed by a small region of internal compression. This mixed compression configuration can balance the problems of high external drag in the case of full external compression and excessive viscous effects during full internal compression. The optimal cowl inlet configuration is the well-known “shock-on-lip’’ condition shown in Figure

Optimal cowl configuration with shocks converging on the cowl lip.

During off-nominal flight conditions when the flow Mach number is less than the design value, the shocks will move ahead of the cowl lip and some of the compressed air will escape the inlet resulting in “spillage’’ and a decrease in the mass capture. In flight conditions where the flow Mach number is greater than the design value, the shocks move into the inlet causing multiple reflected shocks, loss of total pressure, possible boundary layer separation, and engine unstart [

The second ionization scenario was presented by Shneider et al. [

Application of magnetic field and ionized region.

Moving magnetic field and stationary ionized zone

Moving magnetic field and coincident ionizing e-beam

Also, it has been independently shown in works such as in [

The equations of motion that govern inviscid, compressible fluid flow in a region are given by Euler's equations [

These equations can be recast into the following form in order to facilitate the implementation of a flux splitting flow solver method used in this study. The equations are limited to 2D and are consistent with the formulations employed for the study of inlets:

In modeling the source terms we consider the effects of a charged flow through the inlet with an applied electromagnetic field. This can be accomplished with the addition of appropriate electrodynamic terms [

This suggests that the conductivity in MHD flows is very low and therefore the current and hence the induced electric field are also very small. This allows us to assume

As a consequence of the above we may introduce a scalar electric potential

With the application of a magnetic field to a charged flow the body forces and volumetric heating effects are no longer negligible. The body force term known as the Lorentz force is given by the vector

In (

The left-hand side of (

Thus, the source terms are then trivially found as follows without the need for a Poisson solver:

We employ the flux splitting method developed by Liou and Steffen [

Similarly for the pressure terms,

The grid chosen for this study is a simple algebraic style grid with a Thomas Middlecoff control function applied for smoothing. The algebraic grid utilizes uniformly spaced grid points in the

In setting the boundary conditions for these simulations we must consider that the equations are hyperbolic in nature for Mach numbers greater than one. We therefore set the inlet conditions equal to those of the free stream and at the exit we use conditions from the interior. Wall boundary conditions on the upper and lower surfaces are reflecting surfaces where the flow normal to the surface is mirrored.

As mentioned earlier, in this study the design variables are the ionization beam angle

Due to the high computing time of the optimization code we first ran a few broad parametric studies to establish the trends of each ionization method followed by the geometry changes. Once we had a general region for an optimal configuration we implemented the Broyden-Fletcher-Goldfarb-Shanno (BFGS) method in the form of the “fmincon’’ function in MATLAB (MATLAB/SIMULINK is the trademark of The MathWorks Inc; Natick, MA,USA) with initial conditions close to the values obtained via the parametric studies. Note this step could be automated via a genetic algorithm-like approach that characterizes the feasible domain of parameters. The function “fmincon’’ is designed to find the minimum of a constrained nonlinear multivariable function [

In our study, the function “fmincon’’ was found to be very sensitive to the step size as well as the scaling of the objective function itself. Several combinations of the above mentioned variables were tried in order to obtain the best results and it was found that a scaling of

The implementation of this optimization routine is laid out in Figure

Validation studies included validating the AUSM scheme and comparing the geometric optimization of the double ramp inlet to results published in the literature. A symmetrical two ramp inlet, represented in Figure

Full inlet demonstrating initial conditions—Shock-canceled case.

Initial inlet configuration with shock canceling

Mach contours for the shock canceled configuration

In previous work by the authors [

This symmetric inlet was optimized for the mass-weighted total pressure loss from free stream to the throat. The optimal geometry of the inlet allows for the shocks to reflect onto the opposing side of the throat. For operation at speed greater or less than design speed the pressure losses will increase. The initial configuration is the shock-canceled case where the shocks reflect at the centerline and converge on to the throat. The initial ramp angles do not represent an optimal configuration for any Mach number; they are simply a nonoptimal case for our given Mach number.

Figure

Shock-canceled inlet flow conditions.

Region | Mach number | Flow angle | ||
---|---|---|---|---|

1 | 14.000 | 0.0 | 1 | 1 |

2 | 12.107 | 2.882 | 0.927 | 0.927 |

3 | 8.953 | 9.342 | 0.669 | 0.620 |

4 | 6.385 | 0.0 | 0.626 | 0.388 |

The early version of the code was validated in our previous work [

We note that the results of this validation, given in Table

Summary of initial and final inlet conditions.

Pressure recovery | |||
---|---|---|---|

Initial condition | |||

Optimized value | |||

Korte optimized values | 4.263 | 7.621 | 0.625 |

Final optimized inlet configuration Mach number contours.

Buoyed by the confidence in the optimization procedure and the flow solution after validation against published results (see earlier section), the procedure was applied to simultaneous optimization of the ionization beam angle and the ramp geometry. This section summarizes the results obtained for the Cowl style inlet (the main focus of the study). The results of these trials are shown in Figure

Mach contours of optimized cowl inlet.

To simulate a less than design Mach number flow we adjust the ramp angles to

Mach contours for off nominal cowl inlet.

Given this off-nominal design condition, we investigate the ability of an applied magnetic field to direct the flow back to the optimal mass capture configuration. Two different scenarios are considered: (

Figures

Mass capture for inlet with conductivity

Coincident e-beam ionization

Stationary ionization

Mass capture for inlet with conductivity

Coincident e-beam ionization

Stationary ionization

Mass capture for inlet with conductivity

Coincident e-beam ionization

Stationary ionization

Mass capture for inlet with conductivity

Coincident e-beam ionization

Stationary ionization

As mentioned before, based on the results from a broad parametric study, the problem and the optimization procedure was set up. To account for the sensitivity of the optimization routine several different initial conditions were evaluated to see how well the results converged. Finally, the initial conditions for the angle

Table of optimized magnetic field angle for coincident ionized zone.

Initial condition (deg) | ||

Optimized value (deg) |

Table of optimized magnetic field angle for stationary ionized zone.

Initial condition (deg) | ||

Optimized value (deg) |

Comparing the results of the optimizer with those of the parametric study for the stationary ionized zone we can see a very nice correlation. The spread in the optimized results is not large and is consistent with the parametric study. However in comparing the optimizer results with those of the parametric study in the case of the coincident ionization zone we see a bit of discrepancy. There is a large spread in the values given by the optimizer depending on whether we began the optimization above or below the peak value shown in the parametric study.

It is to be mentioned that a limitation of the current study is that it ignores the specific effects of power deposition due to the electron beam. The optimization problem was solved with the intent of finding a solution for the most optimistic scenario and as a result the power deposition was ignored to keep the simulation complexity low. To model the aspect of power deposition [

In order to better understand this result we conducted another parametric study around the peak value. We limited the study to the

Zoom in of peak values for the moving ionization zone.

Finally, Figure

Optimization design history for (a) the magnetic field angle and (b) geometry angles.

Magnetic field angle history

Geometry angle history

Flowchart of optimization routine.

Figure

The final configuration for the geometry is very close to that of the optimal mass capture with no MHD source term present (see Figure

Table of optimized geometry and magnetic field angles for stationary ionized zone.

Initial condition | |||

Optimized value |

This paper studied the numerical optimization of a double ramp cowl style scramjet inlet using magnetohydrodynamic (MHD) effects together with inlet ramp angle changes. An AUSM-based flow solver was utilized to solve the 2D inviscid, compressible Euler equations subject to MHD source terms. The objective function in the optimization was the mass capture at the throat of a cowl style inlet, so that spillage effects for less than design Mach numbers are reduced. The optimization procedure implemented in this study was a numerical Broyden-Flecher-Goldfarb-Shanno- (BFGS-) based procedure. Numerical validation results compared to published results in the literature have been summarized at various stages that include flow solution and the optimization procedure. It is shown that spillage occurring from off-nominal geometries can be reduced by employing MHD control. We also demonstrate a more attractive case of spillage reduction employing simultaneous optimization of the ionization beam angle and the ramp angles.

Ramp angles (degrees)

Magnetic field strength (Tesla)

Direction of the e-beam measured from

Fluid density (

Fluid momenta in

pressure (

Energy (Joules)

Velocity (

External force (N)

Flow solution vector

Flux vectors

Source term

Ratio of specific heats

Temperature (

Gas constant (

Speed of sound (

Magnetic Reynolds number

Permeability of free space (

Gas conductivity (

Characteristic length (m)

Electric field intensity (

Electric potential (

Current density

Total pressure at the

Objective function

Lagrange multipliers

Constraint function.

The authors gratefully acknowledge HyperComp Inc. (POC. Dr. Ramakanth Munipalli) for the financial and technical support for this work.