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Modeling
This page discusses some aspects of flow modeling using MachLine.
Wakes are sheets of vorticity shed aft of lifting surfaces. In potential flow, wakes are sheets of varying doublet strength (vorticity) representing the inviscid limit of the true viscous wake shed by the lifting surface. This shed vorticity is responsible for downwash on the lifting surface, which reduces lift and produces induced drag.
A doublet panel induces a jump in velocity potential between its two faces. This means that the velocity potential is discontinuous across the wake sheet. As such, the velocity potential on the outer surface of the body is discontinuous where the wake is shed from the body, typically sharp trailing edges. As the inner potential is continuous at such intersections, it is necessary for there to be a discontinuity in doublet strength at these intersections. At edges which shed wake panels, the strength of these wake panels is determined by the difference in doublet strength between the two panels forming that edge. This ensures that a line vortex is not formed at wake-shedding edges, even though the doublet strength on the body is discontinuous.
Within MachLine, wakes are generated automatically, but the user has some freedom in specifying how this should be done. For example, if the user desires to perform a nonlifting analysis, then they may specify that there be no edges on the body at which the doublet strength is discontinuous and no wake panels shed. The user may also specify the maximum flow-turning angle which is allowed on the surface before a wake is shed.
In supersonic flows, influences do not propagate upstream. Thus, in many cases, it is unnecessary to explicitly model the wake (for example, on a single straight wing with supersonic trailing edges). In this case, however, the doublet strength still must be discontinuous along edges from which the wake is shed, even though the wake itself is not present. This may be affected by specifying in the input file that a wake is present but it is not to be appended.
How exactly MachLine models the wake is set in the input file, described here.
There are many formulations for enforcing the zero-normal-mass-flux boundary condition in panel methods. Two have been implemented in MachLine: the Morino and source-free formulations. The formulation is specified in the input file, described here.
With the Morino formulation, the inner perturbation potential is chosen to be zero. From this, the source strengths are calculated explicitly as a function of the freestream velocity to satisfy the zero mass-flux boundary condition. A set of control points is then placed within the body and the doublet strengths are solved for such that the perturbation potential at each of these control points is zero.
With the source-free formulation, the inner total potential is chosen to be zero. The choice means that the source strengths are all zero, meaning source contributions never have to be calculated. As with the Morino formulation, a set of control points is placed within the body and the doublet strengths are solved for such that the total potential inside the configuration will result in zero mass flux on the boundary.
Application of the Morino or source-free formulation results in the formation of a linear system of equations, the solution of which entirely specifies the flow about the configuration. This linear system may be solved using a variety of matrix solvers. Those implemented in MachLine are described here. Broadly, we may subdivide them into direct and iterative solvers.
MachLine has two direct solvers built in: LU decomposition ("LU" in the input) and Purcell's method ("PURC" in the input). Both use partial pivoting for numerical stability. Direct solvers provide an "exact" solution to the linear system of equations. However, they do so at the expense of computation time. Both LU decomposition and Purcell's method have O(N^3) complexity, making them particularly time-consuming for meshes with large numbers of panels.
MachLine has two iterative solvers built in: block-Jacobi ("BJAC" in the input) and block-symmetric-successive-overrelaxation ("BSOR" in the input). These are standard block-iterative methods which provide an approximate solution to the linear system of equations. However, for large systems of equations, they are much faster than direct methods.
In testing MachLine, we have found that these iterative solvers perform well for nonlifting flows (i.e. no wake). However, for flows where wakes are present, they fail to converge to an acceptable level of error. As such, we recommend caution when using these iterative methods.
MachLine is capable of calculating surface pressures using various formulas. This is because of the many different ways in which Bernoulli's equation may be written and approximated. The available rules are as follow:
| Rule | Description |
|---|---|
| Incompressible | This is the classic, exact pressure coefficient formula based on the assumption that the freestream Mach number is zero. It may not be selected if the freestream Mach number is set to greater than zero. |
| Isentropic | This is another exact formula, except it allows for a non-zero Mach number. It may not be selected if the freestream Mach number is set to zero. |
| Second-Order | This pressure rule is based on the quadratic approximation to Bernoulli's equation. It may be selected for all freestream Mach numbers. |
| Slender-Body | This is the same as the second-order rule, except that the square of the x perturbation velocity is assumed to be negligible. It may be selected for all freestream Mach numbers. |
| Linear | This rule drops all nonlinear terms from the second-order rule. It may be selected for all freestream Mach numbers. |
Several subsonic pressure corrections are also implemented in MachLine. The current options available are the Prandtl-Glauert, Karman-Tsien, and Laitone corrections. These corrections all solve the flow as being incompressible (freestream Mach number is zero) and then alter the results afterwards by applying correction factors. These correction factors implement a coordinate transformation using a 'correction Mach number' and the incompressible pressure coefficient solutions. You can select any and all of these pressure corrections and MachLine will output the results from each selected pressure correction to the identified location. Note that we do not consider these pressure corrections to be as accurate as modeling the subsonic flow directly and calculating the pressure coefficient using one of the rules described above.