Incident shock waves

Introduction

This tutorial presents the solution of the Rankine–Hugoniot equations for a planar incident shock wave in a supersonic flow. A schematic is shown in Fig. 4. The shock wave is characterized by the upstream velocity \(u_1\) and thermodynamic state, which for an ideal gas is defined by its chemical composition (\(\boldsymbol{n}_1\)) and two independent state variables (e.g., temperature \(T_1\) and pressure \(p_1\)). Across the shock, the flow undergoes an abrupt and irreversible change in state: the downstream properties \((u_2, p_2, T_2)\) follow from the conservation of mass, momentum, and energy, while the post-shock composition \(\boldsymbol{n}_2\) is obtained according to the selected chemical model—either frozen or chemical equilibrium—applied at the downstream state.

Fig. 4 Schematic of a planar shock wave in the wave-fixed frame.

Governing equations

For a one-dimensional planar shock, the Rankine–Hugoniot relations express the conservation of mass, momentum, and energy across the shock front as:

\begin{equation} p_2 = p_1 + \rho_1 u_1^2 \left( 1-\dfrac{\rho_1}{\rho_2}\right) \quad \text{and} \quad h_2 = h_1 + \dfrac{u_1^2}{2}\left[1- \left(\dfrac{\rho_1}{\rho_2}\right)^2\right], \end{equation}

where \(h\) denotes specific enthalpy. This equation must be supplemented by the equation of state, which for an ideal gas reads

\begin{equation} p = \rho R T / W, \end{equation}

where \(R\) is the universal gas constant and \(W\) is the molecular weight of the gas.

Numerical example

We now illustrate how to solve the Rankine–Hugoniot equations in the Combustion Toolbox using the class ShockSolver() class, part of the +combustiontoolbox.+shockdetonation (CT-SD) subpackage (module). Below is an example that solves the Rankine-Hugoniot equations for a planar incident shock wave in air (79% \(\text{N}_2\), 21% \(\text{O}_2\) by volume), with an initial temperature \(T_1 = 300\) K, pressure \(p_1 = 1\) bar, and a pre-shock Mach number \(\mathcal{M}_1 \in [1, 10]\):

% Import packages
import combustiontoolbox.databases.NasaDatabase
import combustiontoolbox.core.*
import combustiontoolbox.shockdetonation.ShockSolver

% Get NASA's database
DB = NasaDatabase();

% Define chemical system
system = ChemicalSystem(DB);

% Initialize mixture
mix = Mixture(system);

% Define chemical state
set(mix, {'N2', 'O2'}, [79/21, 1]);

% Define properties
mixArray1 = setProperties(mix, 'temperature', 300, 'pressure', 1, 'Mach', 1:0.1:10);

% Initialize shock solver
solver = ShockSolver();

% Perform shock calculations
[mixArray1, mixArray2] = solveArray(solver, mixArray1);

% Generate report
report(solver, mixArray1, mixArray2);

This code snippet will generate two figures: Molar fraction of species in the mixture as a function of the pre-shock Mach number and the variation of the thermodynamic properties (e.g., temperature, pressure) as a function of the pre-shock Mach number.

Fig. 5 Molar fraction of chemical species downstream of a normal shock in air (79% \(\text{N}_2\), 21% \(\text{O}_2\) by volume) as a function of the pre-shock Mach number \(\mathcal{M}_1\), for standard upstream conditions \(T_1 = 300\) K, \(p_1 = 1\) bar.

Fig. 6 Thermodynamic properties downstream of a normal shock in air (79% \(\text{N}_2\), 21% \(\text{O}_2\) by volume) as function of the pre-shock Mach number \(\mathcal{M}_1\), for standard upstream conditions \(T_1 = 300\) K, \(p_1 = 1\) bar.

Tip

The Combustion Toolbox supports multiple caloric models for the final gas mixture through the CaloricGasModel() class. By default, CT uses the calorically imperfect gas model. To specify a different caloric model, set the property caloricGasModel when initializing the ShockSolver() class.

For example, to consider the calorically perfect gas approximation, initialize the ShockSolver() class as follows

solver = ShockSolver('problemType', 'SHOCK_I', 'caloricGasModel', CaloricGasModel.perfect);

For the thermally perfect gas model, also known as the calorically imperfect gas with frozen chemistry model, use

solver = ShockSolver('problemType', 'SHOCK_I', 'caloricGasModel', CaloricGasModel.thermallyPerfect);