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SIMULATION OF 1-D SUPERSONIC NOZZLE FLOW USING MAC-CORMACK METHOD

PROBLEM SETUP DESCRIPTION 1. Steady isentropic flow through a Convergent Divergent Nozzle is Assumed. 2. Source flow into the inlet of the nozzle is at constant temperature and pressure. 3. Isentropic supersonic expansion of the flow occurs as denoted by the Mach Number Values as shown is the above schematic.…

MATLAB

Shouvik Bandopadhyay
updated on 25 Apr 2019

PROBLEM SETUP DESCRIPTION

$\"\"$

1. Steady isentropic flow through a Convergent Divergent Nozzle is Assumed.

2. Source flow into the inlet of the nozzle is at constant temperature and pressure.

3. Isentropic supersonic expansion of the flow occurs as denoted by the Mach Number Values

as shown is the above schematic.

4. It is assumed that flow field varies only in the x direction. The said flow is known

as quasi-1D flow.

Non Dimensional Governing Equations for Conservative Form Analysis

1. Continuity Equation:

$\frac{δ (ρ' . A')}{δ t'} + \frac{δ (ρ' . A' . v')}{δ x'} = 0$ $(delta(rho\'.A\'))/(deltat\')+(delta(rho\'.A\'.v\'))/(deltax\')=0$

where

$ρ' = \frac{ρ}{ρ_{o}}$ $rho\' = rho/rho_o$ $A' = \frac{A}{A_{*}}$ $A\'=A/(A_ast)$ $t' = \frac{t}{(\frac{L}{a_{0}})}$ $t\'=t/((L/a_0))$ $v' = \frac{v}{a_{0}}$ $v\'=v/a_0$ $x' = \frac{x}{L}$ $x\'=x/L$

$ρ_{0} \to$ $rho_0rarr$ Density of reservoir $L \to$ $Lrarr$ Length of the nozzle

$a_{0} \to$ $a_0rarr$ Sound Speed inside the reservoir $A_{*} \to$ $A_astrarr$ Throat Area

$ρ \to$ $rhorarr$ Density of working fluid $v \to$ $vrarr$ Velocity of Working FLuid

$A \to$ $Ararr$ Area of Nozzle

2. Momentum Equation

$\frac{δ (ρ' . A' . v')}{δ t'} + \frac{δ (ρ' . A' . v'^{2} + (\frac{1}{γ}) ρ' . A')}{δ x'} = (\frac{1}{γ}) . p' . \frac{δ A'}{δ x'}$ $(delta(rho\'.A\'.v\'))/(deltat\')+(delta(rho\'.A\'.v\' ^2+(1/gamma)rho\'.A\'))/(deltax\')=(1/gamma).p\'.(deltaA\')/(deltax\')$

where

$p' = \frac{p}{p_{0}}$ $p\'=p/p_0$ $γ \to$ $gammararr$ Specific Heat Ratio`

$p_{0} \to$ $p_0rarr$ Pressure of reservoir $p \to$ $prarr$ Fluid Pressure

3. Energy Equation

$δ \frac{(ρ' . (\frac{e'}{γ - 1} + \frac{γ}{2} . v'^{2}) . A')}{δ t'} + \frac{δ (ρ' . (\frac{e'}{γ - 1} + \frac{γ}{2} . v'^{2}) . v' . A' + p' . A' . v')}{δ x'} = 0$ $delta((rho\'.((e\')/(gamma-1)+gamma/2.v\'^2).A\'))/(deltat\')+(delta(rho\'.((e\')/(gamma-1)+gamma/2.v\'^2).v\'.A\'+p\'.A\'.v\'))/(deltax\')=0$

where

$e' = \frac{e}{e_{0}}$ $e\'=e/e_0$

$e_{0} \to$ $e_0rarr$ Energy of reservoir

T = temperarure

e\' = T\'

$C_{v} \to$ $C_vrarr$ Specific Heat at Constant Volume

Let U be the solutions vector, F be the flux vector and J be the source term, such that: -

$U_{1} = ρ' A'$ $U_1 = rho\' A\'$

$U_{2} = ρ' A' V'$ $U_2 = rho\' A\' V\'$

$U_{3} = ρ' (\frac{e'}{γ - 1} + \frac{γ}{2} V'^{2}) A'$ $U_3 = rho\'((e\')/(gamma-1)+gamma/2 V\'^2)A\'$

$F_{1} = ρ' A' V'$ $F_1 = rho\' A\' V\'$

`F_2 = rho\' A\' V\'^2 + 1/gamma p\'A\'

$F_{3} = ρ' (\frac{e'}{γ - 1} + \frac{γ}{2} V'^{2}) V' A' + p' A' V'$ $F_3 = rho\'((e\')/(gamma-1) + gamma/2 V\'^2)V\' A\' + p\' A\' V\'$

$J_{2} = \frac{1}{γ} p' \frac{\partial A'}{\partial x'}$ $J_2 = 1/gamma p\' (del A\')/(del x\')$

Substituting the above terms in the governing equations (conservative form) we get:

$\frac{\partial U_{1}}{\partial t'} = - \frac{\partial F_{1}}{\partial x'}$ $(del U_1)/(del t\') = -(del F_1)/(del x\')$

$\frac{\partial U_{2}}{\partial t'} = J 2 - \frac{\partial F_{2}}{\partial x'}$ $(del U_2)/(del t\') = J2 - (del F_2)/(del x\')$

$\frac{\partial U_{3}}{\partial t'} = - \frac{\partial F_{3}}{\partial x'}$ $(del U_3)/(del t\') = -(del F_3)/(del x\')$

The solver built in MATLAB solves for the solution vectors. The following equations are utilized to caliberate the corrosponding parameters.

$ρ' = \frac{U'}{A'}$ $rho\' = (U\')/(A\')$

$V' = \frac{U_{2}}{U_{1}}$ $V\' = U_2/U_1$

$T' = e' = (γ - 1) (\frac{U_{3}}{U_{1}} - \frac{γ}{2} V'^{2})$ $T\' = e\' = (gamma-1)(U_3/U_1-gamma/2 V\'^2)$

$p' = ρ' T'$ $p\' = rho\' T\'$

As we\'re solving for the solution vectors, the RHS consisting of flux terms and source terms must also be expressed in terms of U1, U2 and U3 to obtain the pure form of the flux terms. Otherwise, the solution may be rendered unstable after just few time steps.

$F_{1} = U_{2}$ $F_1 = U_2$

$F_{2} = \frac{U_{2}^{2}}{U_{1}} + \frac{γ - 1}{γ} (U_{3} - \frac{γ}{2} \frac{U_{2}^{2}}{U_{1}})$ $F_2 = U_2^2/U_1 + (gamma-1)/gamma(U_3 - gamma/2 U_2^2/U_1)$

$F_{3} = γ \frac{U_{2} U_{3}}{U_{1}} - \frac{γ (γ - 1)}{2} \frac{U_{2}^{3}}{U_{1}^{2}}$ $F_3 = gamma (U_2 U_3)/U_1 - (gamma(gamma-1))/2 U_2^3/U_1^2$

$J_{2} = \frac{γ - 1}{γ} (U_{3} - \frac{γ}{2} \frac{U_{2}^{2}}{U_{1}}) \frac{\partial (ln A')}{\partial x'}$ $J_2 = (gamma-1)/gamma(U_3 - gamma/2 U_2^2/U_1)(del(ln A\'))/(del x\')$

NON DIMENSIONAL GOVERNING EQUATIONS FOR NON-CONSERVATIVE FORM ANALYSIS

NOTE: Symbols have usual meanings as before.

1. Continuity Equation

$\frac{δ ρ'}{δ t'} = - ρ' . \frac{δ v'}{δ x'} - ρ' . v' \frac{δ (ln A')}{δ x'} - v' . \frac{δ ρ'}{δ x'}$ $(deltarho\')/(deltat\')=-rho\'.(deltav\')/(deltax\')-rho\'.v\'(delta(lnA\'))/(deltax\')-v\'.(deltarho\')/(deltax\')$

2. Momentum Equation

$\frac{δ v'}{δ t'} = - v' . \frac{δ v'}{δ x'} - (\frac{1}{γ}) [\frac{δ T'}{δ x'} + (\frac{T'}{ρ'}) . \frac{δ p'}{δ x'}]$ $(deltav\')/(deltat\')=-v\'.(deltav\')/(deltax\')-(1/gamma)[(deltaT\')/(deltax\')+((T\')/(rho\')).(deltap\')/(deltax\')]$

3. Energy Equation

$\frac{δ T'}{δ t'} = - v' . \frac{δ T'}{δ x'} - (γ - 1) . T' [\frac{δ v'}{δ x'} + v' . \frac{δ (ln (A'))}{δ x'}]$ $(deltaT\')/(deltat\')=-v\'.(deltaT\')/(deltax\')-(gamma-1).T\'[(deltav\')/(deltax\')+v\'.(delta(ln(A\')))/(deltax\')]$

DISCRETIZATION OF THE NON-CONSERVATIVE FORM OF GOVERNING EQUATION

MacCormack\'s Predictor Corrector Method is used to discretize the governing equations.

Predictor Step

Computing time derivatives using forward difference scheme for space derivatives by substituting known values of flow variables at time t.

${(\frac{\partial ρ}{\partial t})}_{i}^{t} = - ρ_{i}^{t} [\frac{V_{i + 1}^{t} - V_{i}^{t}}{Δ x}] - ρ_{i}^{t} V_{i}^{t} [\frac{ln (A_{i + 1}) - ln (A_{i})}{Δ x}] - V_{i}^{t} [\frac{ρ_{i + 1}^{t} - ρ_{i}^{t}}{Δ x}]$ $((del rho)/(del t))_i^t = -rho_i^t[(V_(i+1)^t - V_i^t)/(Delta x)] - rho_i^t V_i^t[(ln(A_(i+1)) - ln(A_i))/(Delta x)] - V_i^t[(rho_(i+1)^t - rho_i^t)/(Delta x)]$

${(\frac{\partial V}{\partial t})}_{i}^{t} = - V_{i}^{t} [\frac{V_{i + 1}^{t} - V_{i}^{t}}{Δ x}] - \frac{1}{γ} [\frac{T_{i + 1}^{t} - T_{i}^{t}}{Δ x} + \frac{T_{i}^{t}}{ρ_{i}^{t}} (\frac{ρ_{i + 1}^{t} - ρ_{i}^{t}}{Δ x})]$ $((del V)/(del t))_i^t = -V_i^t[(V_(i+1)^t - V_i^t)/(Delta x)] - 1/gamma[(T_(i+1)^t - T_i^t)/(Delta x) + T_i^t/rho_i^t( (rho_(i+1)^t - rho_i^t)/(Delta x))]$

${(\frac{\partial T}{\partial x})}_{i}^{t} = - V_{i}^{t} [\frac{T_{i + 1}^{t} - T_{i}^{t}}{Δ x}] - (γ - 1) T_{i}^{t} [\frac{V_{i + 1}^{t} - V_{i}^{t}}{Δ x} + V_{i}^{t} (\frac{ln (A_{i + 1}) - ln (A_{i})}{Δ x})]$ $((del T)/(del x))_i^t = -V_i^t[(T_(i+1)^t - T_i^t)/(Delta x)] - (gamma-1)T_i^t [(V_(i+1)^t-V_i^t)/(Delta x)+ V_i^t((ln(A_(i+1)) - ln(A_i))/(Delta x))]$

Computing the predicted values of flow field variables at time (t+dt) using the time derivatives at time t.

${(¯ ρ)}_{i}^{t + Δ t} = ρ_{i}^{t} + {(\frac{\partial ρ}{\partial t})}_{i}^{t} Δ t$ $(bar rho)_i^(t+Delta t) = rho_i^t + ((del rho)/(del t))_i^t Delta t$

${(¯ ¯ ¯ V)}_{i}^{t + Δ t} = V_{i}^{t} + {(\frac{\partial V}{\partial t})}_{i}^{t} Δ t$ $(bar V)_i^(t+Delta t) = V_i^t + ((del V)/(del t))_i^t Delta t$

${(¯ ¯ ¯ T)}_{i}^{t + Δ t} = T_{i}^{t} + {(\frac{\partial T}{\partial t})}_{i}^{t} Δ t$ $(bar T)_i^(t+Delta t) = T_i^t + ((del T)/(del t))_i^t Delta t$

where dt is the time step size satisfying the CFL condition (with C = 0.5)

$Δ t = C (\frac{Δ x}{\sqrt{T} + V})$ $Delta t = C((Delta x)/(sqrt(T)+V))$

The time step size is calculated for all nodes and the minimum of all time step size values is takes as the value for \'dt\'.

Corrector Step

Computing time derivative values at time (t+dt) with the predicted values of flow variables.

${(¯ ¯¯¯¯¯ ¯ \frac{\partial ρ}{\partial t})}_{i}^{t + Δ t} = - {(¯ ρ)}_{i}^{t + Δ t} ⎡ ⎢ ⎣ \frac{{(¯ ¯ ¯ V)}_{i}^{t + Δ t} - {(¯ ¯ ¯ V)}_{i - 1}^{t + Δ t}}{Δ x} ⎤ ⎥ ⎦ - {(¯ ρ)}_{i}^{t + Δ t} {(¯ ¯ ¯ V)}_{i}^{t + Δ t} [\frac{ln (A_{i}) - ln (A_{i - 1})}{Δ x}] - {(¯ ¯ ¯ V)}_{i}^{t + Δ t} [\frac{{(¯ ρ)}_{i}^{t + Δ t} - {(¯ ρ)}_{i - 1}^{t + Δ t}}{Δ x}]$ $(bar((del rho)/(del t)))_i^(t+Delta t) = -(bar rho)_i^(t+Delta t)[((bar V)_i^(t+ Delta t) - (bar V)_(i-1)^(t+Delta t))/(Delta x)] - (bar rho)_i^(t+Delta t)(bar V)_i^(t+Delta t)[(ln(A_i) - ln(A_(i-1)))/(Delta x)] - (bar V)_i^(t+Delta t)[((bar rho)_i^(t+Delta t) - (bar rho)_(i-1)^(t+Delta t))/(Delta x)]$

${(¯ ¯¯¯¯¯¯ ¯ \frac{\partial V}{\partial t})}_{i}^{t + Δ t} = - {(¯ ¯ ¯ V)}_{i}^{t + Δ t} ⎡ ⎢ ⎣ \frac{{(¯ ¯ ¯ V)}_{i}^{t + Δ t} - {(¯ ¯ ¯ V)}_{i - 1}^{t + Δ t}}{Δ x} ⎤ ⎥ ⎦ - \frac{1}{γ} ⎡ ⎢ ⎣ \frac{{(¯ ¯ ¯ T)}_{i}^{t + Δ t} - {(¯ ¯ ¯ T)}_{i - 1}^{t + Δ t}}{Δ x} + \frac{{(¯ ¯ ¯ T)}_{i}^{t + Δ t}}{{(¯ ρ)}_{i}^{t + Δ t}} (\frac{{(¯ ρ)}_{i}^{t + Δ t} - {(¯ ρ)}_{i - 1}^{t + Δ t}}{Δ x}) ⎤ ⎥ ⎦$ $(bar((del V)/(del t)))_i^(t+ Delta t) = -(bar V)_i^(t+Delta t)[((bar V)_i^(t+Delta t) - (bar V)_(i-1)^(t+Delta t))/(Delta x)] - 1/gamma[((bar T)_i^(t+Delta t) - (bar T)_(i-1)^(t+Delta t))/(Delta x) + (bar T)_i^(t+Delta t)/(bar rho)_i^(t+Delta t)( ((bar rho)_i^(t+Delta t) - (bar rho)_(i-1)^(t+Delta t))/(Delta x))]$

${(¯ ¯¯¯¯¯¯ ¯ \frac{\partial T}{\partial x})}_{i}^{t + Δ t} = - {(¯ ¯ ¯ V)}_{i}^{t + Δ t} ⎡ ⎢ ⎣ \frac{{(¯ ¯ ¯ T)}_{i}^{t + Δ t} - {(¯ ¯ ¯ T)}_{i - 1}^{t + Δ t}}{Δ x} ⎤ ⎥ ⎦ - (γ - 1) T_{i}^{t + Δ t} ⎡ ⎢ ⎣ \frac{{(¯ ¯ ¯ V)}_{i}^{t + Δ t} - {(¯ ¯ ¯ V)}_{i - 1}^{t + Δ t}}{Δ x} + {(¯ ¯ ¯ V)}_{i}^{t + Δ t} (\frac{ln (A_{i}) - ln (A_{i - 1})}{Δ x}) ⎤ ⎥ ⎦$ $(bar((del T)/(del x)))_i^(t+Delta t) = -(bar V)_i^(t+Delta t)[((bar T)_i^(t+Delta t) - (bar T)_(i-1)^(t+Delta t))/(Delta x)] - (gamma-1)T_i^(t+Delta t) [((bar V)_i^(t+Delta t)- (bar V)_(i-1)^(t+Delta t))/(Delta x)+ (bar V)_i^(t+Delta t)((ln(A_i) - ln(A_(i-1)))/(Delta x))]$

Calculating the average of time derivative values obtained from foward difference and backward difference.

${(\frac{\partial ρ}{\partial t})}_{a v} = \frac{1}{2} ⎡ ⎣ {(\frac{\partial ρ}{\partial t})}_{i}^{t} + {(¯ ¯¯¯¯¯ ¯ \frac{\partial ρ}{\partial t})}_{i}^{t + Δ t} ⎤ ⎦$ $((del rho)/(del t))_(av) = 1/2[((del rho)/(del t))_i^t + (bar((del rho)/(del t)))_i^(t+Delta t)]$

${(\frac{\partial V}{\partial t})}_{a v} = \frac{1}{2} [{(\frac{\partial V}{\partial t})}_{i}^{t} + {(¯ ¯¯¯¯¯¯ ¯ \frac{\partial V}{\partial t})}_{i}^{t + Δ t}]$ $((del V)/(del t))_(av) = 1/2[((del V)/(del t))_i^t + (bar((del V)/(del t)))_i^(t+Delta t)]$

${(\frac{\partial T}{\partial t})}_{a v} = \frac{1}{2} [{(\frac{\partial T}{\partial t})}_{i}^{t} + {(¯ ¯¯¯¯¯¯ ¯ \frac{\partial T}{\partial t})}_{i}^{t + Δ t}]$ $((del T)/(del t))_(av) = 1/2[((del T)/(del t))_i^t + (bar((del T)/(del t)))_i^(t+Delta t)]$

Compute the corrected values of flow field variables using the average value of time derivatives.

$ρ_{i}^{t + Δ t} = ρ_{i}^{t} + {(\frac{\partial ρ}{\partial t})}_{a v} Δ t$ $rho_i^(t+Delta t) = rho_i^t + ((del rho)/(del t))_(av) Delta t$

$V_{i}^{t + Δ t} = V_{i}^{t} + {(\frac{\partial V}{\partial t})}_{a v} Δ t$ $V_i^(t+Delta t) = V_i^t + ((del V)/(del t))_(av) Delta t$

$T_{i}^{t + Δ t} = T_{i}^{t} + {(\frac{\partial T}{\partial t})}_{a v} Δ t$ $T_i^(t+Delta t) = T_i^t + ((del T)/(del t))_(av) Delta t$

$p_{i}^{t + Δ t} = ρ_{i}^{t + Δ t} . T_{i}^{t + Δ t}$ $p_i^(t+Delta t) = rho_i^(t+Delta t) .T_i^(t+Delta t)$

DISCRETIZATION OF THE CONSERVATIVE FORM OF GOVERNING EQUATION

MacCormack\'s Predictor Corrector Method has been used for discretization.

Predictor Step

The flux terms and the source terms are calculated at nodes i and i+1 using previous state values of U1, U2 and U3 and forward difference is applied to compute the value of the time derivatives.

${(\frac{\partial U_{1}}{\partial t'})}_{i}^{t} = - \frac{F_{1 (i + 1)}^{t} - F_{1 (i)}^{t}}{Δ x'}$ $((del U_1)/(del t\'))_i^t = -(F_(1(i+1))^t - F_(1(i))^t)/(Delta x\')$

${(\frac{\partial U_{2}}{\partial t'})}_{i}^{t} = \frac{J_{2 (i + 1)}^{t} - J_{2 (i)}^{t}}{Δ x'} - \frac{F_{2 (i + 1)}^{t} - F_{2 (i)}^{t}}{Δ x'}$ $((del U_2)/(del t\'))_i^t = (J_(2(i+1))^t-J_(2(i))^t)/(Delta x\') - (F_(2(i+1))^t - F_(2(i))^t)/(Delta x\')$

${(\frac{\partial U_{3}}{\partial t'})}_{i}^{t} = - \frac{F_{3 (i + 1)}^{t} - F_{3 (i)}^{t}}{Δ x'}$ $((del U_3)/(del t\'))_i^t = -(F_(3(i+1))^t - F_(3(i))^t)/(Delta x\')$

These time derivative values are used to calculate the predicted values of U1, U2 and U3.

${¯ ¯ ¯ U}_{1 (i)}^{t + Δ t} = {(\frac{\partial U_{1}}{\partial t'})}_{i}^{t} . Δ t'$ $bar(U)_(1(i))^(t+Delta t) = ((del U_1)/(del t\'))_i^t .Delta t\'$

${¯ ¯ ¯ U}_{2 (i)}^{t + Δ t} = {(\frac{\partial U_{2}}{\partial t'})}_{i}^{t} . Δ t'$ $bar(U)_(2(i))^(t+Delta t) = ((del U_2)/(del t\'))_i^t .Delta t\'$

${¯ ¯ ¯ U}_{3 (i)}^{t + Δ t} = {(\frac{\partial U_{3}}{\partial t'})}_{i}^{t} . Δ t'$ $bar(U)_(3(i))^(t+Delta t) = ((del U_3)/(del t\'))_i^t .Delta t\'$

Corrector Step

New set of flux and source term values are calculated using the predicted values of U1, U2 and U3 and backward difference is applied to compute the time derivative values for the corrector step.

${(\frac{¯ ¯¯¯¯¯ ¯ \partial U_{1}}{\partial t'})}_{i}^{t + Δ t} = - \frac{{¯ ¯ ¯ F}_{1 (i)}^{t + Δ t} - {¯ ¯ ¯ F}_{1 (i - 1)}^{t + Δ t}}{Δ x'}$ $(bar(del U_1)/(del t\'))_i^(t+Delta t) = -(bar(F)_(1(i))^(t+Delta t) - bar(F)_(1(i-1))^(t+Delta t))/(Delta x\')$

${(\frac{¯ ¯¯¯¯¯ ¯ \partial U_{2}}{\partial t'})}_{i}^{t + Δ t} = \frac{{¯ ¯ ¯ J}_{2 (i)}^{t + Δ t} - {¯ ¯ ¯ J}_{2 (i - 1)}^{t + Δ t}}{Δ x'} - \frac{{¯ ¯ ¯ F}_{2 (i)}^{t + Δ t} - {¯ ¯ ¯ F}_{2 (i - 1)}^{t + Δ t}}{Δ x'}$ $(bar(del U_2)/(del t\'))_i^(t+Delta t) = (bar(J)_(2(i))^(t+Delta t)-bar(J)_(2(i-1))^(t+Delta t))/(Delta x\') - (bar(F)_(2(i))^(t+Delta t) - bar(F)_(2(i-1))^(t+Delta t))/(Delta x\')$

${(\frac{¯ ¯¯¯¯¯ ¯ \partial U_{3}}{\partial t'})}_{i}^{t + Δ t} = - \frac{{¯ ¯ ¯ F}_{3 (i)}^{t + Δ t} - {¯ ¯ ¯ F}_{3 (i - 1)}^{t + Δ t}}{Δ x'}$ $(bar(del U_3)/(del t\'))_i^(t+Delta t) = -(bar(F)_(3(i))^(t+Delta t) - bar(F)_(3(i-1))^(t+Delta t))/(Delta x\')$