Oblique / Prandtl-Meyer Flow Solver

class pysagas.cfd.oblique_prandtl_meyer.OPM

Oblique shock thoery and Prandtl-Meyer expansion theory flow solver.

This implementation uses oblique shock theory for flow-facing cell elements, and Prandtl-Meyer expansion theory for rearward-facing elements.

Extended Summary

Data attribute ‘method’ refers to which method was used for a particular cell, according to:

• -1 : invalid / skipped (eg. 90 degree face)

• 0 : parallel face, do nothing

• 1 : Prandlt-Meyer

• 2 : normal shock

• 3 : oblique shock

static _load_sens_data(sensitivity_filepath)

Extracts design parameters from the sensitivity data.

Parameters:

sensitivity_filepath (str)

Return type:

Tuple[DataFrame, list[set]]

static _solve_normal(M1, p1=1.0, T1=1.0, gamma=1.4)

Solves the flow using normal shock theory.

Parameters:

• M1 (float) – The pre-shock Mach number.

• p1 (float, optional) – The pre-shock pressure (Pa). The default is 1.0.

• T1 (float, optional) – The pre-expansion temperature (K). The default is 1.0.

• gamma (float, optional) – The ratio of specific heats. The default is 1.4.

Returns:

• M2 (float) – The post-shock Mach number.

• p2 (float) – The post-shock pressure (Pa).

• T2 (float) – The post-shock temperature (K).

static _solve_oblique(theta, M1, p1=1.0, T1=1.0, gamma=1.4)

Solves the flow using oblique shock theory.

Parameters:

• theta (float) – The flow deflection angle, specified in radians.

• M1 (float) – The pre-shock Mach number.

• p1 (float, optional) – The pre-shock pressure (Pa). The default is 1.0.

• T1 (float, optional) – The pre-expansion temperature (K). The default is 1.0.

• gamma (float, optional) – The ratio of specific heats. The default is 1.4.

Returns:

• M2 (float) – The post-shock Mach number.

• p2 (float) – The post-shock pressure (Pa).

• T2 (float) – The post-shock temperature (K).

static _solve_pm(theta, M1, p1=1.0, T1=1.0, gamma=1.4)

Solves for the Mach number, pressure and temperature after a Prandtl-Meyer expansion fan.

Parameters:

• theta (float) – The deflection angle, specified in radians.

• M1 (float) – The pre-expansion Mach number.

• p1 (float, optional) – The pre-expansion pressure (Pa). The default is 1.0.

• T1 (float, optional) – The pre-expansion temperature (K). The default is 1.0.

• gamma (float, optional) – The ratio of specific heats. The default is 1.4.

Returns:

• M2 (float) – The post-expansion Mach number.

• p2 (float) – The post-expansion pressure (Pa).

• T2 (float) – The post-expansion temperature (K).

static beta_max(M, gamma=1.4)

Returns the maximum shock angle for a given Mach number.

Parameters:

• M (float)

• gamma (float)

static dp_dtheta_obl(theta, M1, nominal_flowstate, p1=1.0, T1=1.0, gamma=1.4, perturbation=0.001)

Solves for the sensitivities at the given point using oblique shock theory.

Parameters:

• theta (float) – The deflection angle, specified in radians.

• M1 (float) – The pre-expansion Mach number.

• nominal_flowstate (FlowState) – The nominal flowstate at this point.

• p1 (float, optional) – The pre-expansion pressure (Pa). The default is 1.0.

• T1 (float, optional) – The pre-expansion temperature (K). The default is 1.0.

• gamma (float, optional) – The ratio of specific heats. The default is 1.4.

• perturbation (float, optional) – The perturbation amount. Perturbations are calculated according to a multiple of (1 +/- perturbation). The default is 1e-3.

Returns:

• dM_dtheta (float) – The sensitivity of the post-expansion Mach number with respect to the deflection angle theta.

• dP_dtheta (float) – The sensitivity of the post-expansion pressure (Pa) with respect to the deflection angle theta.

• dT_dtheta (float) – The sensitivity of the post-expansion temperature (K) with respect to the deflection angle theta.

static dp_dtheta_pm(theta, M1, nominal_flowstate, p1=1.0, T1=1.0, gamma=1.4, perturbation=0.001)

Solves for the sensitivities at the given point using Prandtl-Meyer theory.

Parameters:

• theta (float) – The deflection angle, specified in radians.

• M1 (float) – The pre-expansion Mach number.

• nominal_flowstate (FlowState) – The nominal flowstate at this point.

• p1 (float, optional) – The pre-expansion pressure (Pa). The default is 1.0.

• T1 (float, optional) – The pre-expansion temperature (K). The default is 1.0.

• gamma (float, optional) – The ratio of specific heats. The default is 1.4.

• perturbation (float, optional) – The perturbation amount. Perturbations are calculated according to a multiple of (1 +/- perturbation). The default is 1e-3.

Returns:

• dM_dtheta (float) – The sensitivity of the post-expansion Mach number with respect to the deflection angle theta.

• dP_dtheta (float) – The sensitivity of the post-expansion pressure (Pa) with respect to the deflection angle theta.

• dT_dtheta (float) – The sensitivity of the post-expansion temperature (K) with respect to the deflection angle theta.

static inv_pm(angle, gamma=1.4)

Solves the inverse Prandtl-Meyer function using a bisection algorithm.

Parameters:

• angle (float)

• gamma (float)

static oblique_M2(M1, beta, theta, gamma=1.4)

Calculates the Mach number following an oblique shock.

Parameters:

• M1 (float) – The pre-expansion Mach number.

• beta (float) – The shock angle specified in radians.

• theta (float) – The deflection angle, specified in radians.

• gamma (float, optional) – The ratio of specific heats. The default is 1.4.

Returns:

M2 – The post-shock Mach number.

Return type:

float

References

Peter Jacobs

static oblique_T2_T1(M1, beta, gamma=1.4)

Returns the temperature ratio T2/T1 across an oblique shock.

Parameters:

• M1 (float) – The pre-shock Mach number.

• beta (float) – The shock angle specified in radians.

• gamma (float, optional) – The ratio of specific heats. The default is 1.4.

Returns:

T2/T1 – The temperature ratio across the shock.

Return type:

float

References

Peter Jacobs

static oblique_beta(M1, theta, gamma=1.4, tolerance=1e-06)

Calculates the oblique shock angle using a recursive algorithm.

Parameters:

• M1 (float) – The pre-shock Mach number.

• theta (float) – The flow deflection angle, specified in radians.

• gamma (float, optional) – The ratio of specific heats. The default is 1.4.

• tolerance (float)

References

Peter Jacobs

static oblique_p2_p1(M1, beta, gamma=1.4)

Returns the pressure ratio p2/p1 across an oblique shock.

Parameters:

• M1 (float) – The pre-shock Mach number.

• beta (float) – The shock angle specified in radians.

• gamma (float)

Returns:

p2/p1 – The pressure ratio across the shock.

Return type:

float

References

Peter Jacobs

static oblique_rho2_rho1(M1, beta, gamma=1.4)

Returns the density ratio rho2/rho1 across an oblique shock.

Parameters:

• M1 (float) – The pre-shock Mach number.

• beta (float) – The shock angle specified in radians.

• gamma (float, optional) – The ratio of specific heats. The default is 1.4.

Returns:

T2/T1 – The temperature ratio across the shock.

Return type:

float

References

Peter Jacobs

static pm(M, gamma=1.4)

Solves the Prandtl-Meyer function and returns the Prandtl-Meyer angle in radians.

Parameters:

• M (float)

• gamma (float)

save(name)

Save the solution to VTK file format. Note that the geometry mesh must have been loaded from a parser which includes connectivity information (eg. PyMesh or TRI).

Parameters:

• name (str) – The filename prefix of the desired output file.

• attributes (Dict[str, list]) – The attributes dictionary used to initialise the file writer. Note that this is not required when calling solve from a child FlowSolver.

solve(freestream=None, mach=None, aoa=None, cog=Vector(0, 0, 0))

Run the flow solver.

Parameters:

• freestream (Flowstate, optional) – The free-stream flow state. The default is the freestream provided upon instantiation of the solver.

• mach (float, optional) – The free-stream Mach number. The default is that specified in the freestream flow state.

• aoa (float, optional) – The free-stream angle of attack. The default is that specified in the freestream flow state.

• cog (Vector3)

Returns:

result – The flow results.

Return type:

FlowResults

:raises Exception : when no freestream can be found.:

solve_sens(sensitivity_filepath=None, parameters=None, cells_have_sens_data=False, freestream=None, mach=None, aoa=None, cog=Vector(0, 0, 0), perturbation=0.001)

Run the flow solver to obtain sensitivity information.

Parameters:

• freestream (Flowstate, optional) – The free-stream flow state. The default is the freestream provided upon instantiation of the solver.

• Mach (float, optional) – The free-stream Mach number. The default is that specified in the freestream flow state.

• aoa (float, optional) – The free-stream angle of attack. The default is that specified in the freestream flow state.

• sensitivity_filepath (str | None)

• parameters (list[str] | None)

• cells_have_sens_data (bool | None)

• mach (float | None)

• cog (Vector3)

• perturbation (float)

Return type:

SensitivityResults

:raises Exception : when no freestream can be found.:

static theta_from_beta(M1, beta, gamma=1.4)

Calculates the flow deflection angle from the oblique shock angle.

Parameters:

• M1 (float) – The pre-expansion Mach number.

• beta (float) – The shock angle specified in radians.

• gamma (float, optional) – The ratio of specific heats. The default is 1.4.

Returns:

theta – The deflection angle, specified in radians.

Return type:

float

References

Peter Jacobs