# Compressible Flow: de Laval Nozzle

## Overview

The aim of this test case is to validate the following parameters of compressible steady-state turbulent flow through a de Laval Nozzle subsonic and supersonic flow regimes:

• Mach number
• Pressure

The rhoSimpleFoam solver is used for the subsonic case, while for the supersonic case the sonicFoam solver is employed. The kω SST

model was used to model turbulence. Simulation results of SimScale were compared to analytical results obtained from methods elucidated in [1]. The mesh was created locally using the blockMesh tool and then imported on to the SimScale platform.

Import validation project into workspace

## Geometry

A typical configuration of the de Laval nozzle with a non-smooth throat was chosen as the geometry (see Table 1 for coordinates). Since the nozzle is axisymmetric, it was modeled as an angular slice of the complete geometry, with a wedge angle of 18 degrees (see Fig.1.).

Table 1: Point Coordinates
Point x

$x$

y

$y$

z

$z$

A 0 0 0
B 0 5.5434 35
C 0 -5.5434 35
D 68.68 0 0
E 68.68 3.1677 20
F 68.68 -3.1677 20
G 238.8 6.3354 40
H 238.8 -6.3354 40

## Analysis type and Domain

A non-uniformly-spaced hexahedral mesh was generated using the blockMesh tool (see Fig.2.). Flow near the nozzle wall was resolved using inflation of y+=30

${y}^{+}=30$

for the subsonic and 300

$300$

for the supersonic case. In order to keep the flow two-dimensional, the mesh was designed to have only one layer in the y

$y$

direction.

Tool Type : OPENFOAM®

Analysis Type : Compressible Steady-state (Turbulent)

Mesh and Element types :

Table 2: Mesh Metrics
Mesh type Cells in x Cells in y Cells in z Number of nodes Type
blockMesh 150 1 100 15000 2D hex

## Simulation Setup

Fluid:

Table 3 encapsulates the properties of fluids used in the subsonic and supersonic case simulations. The need for using a different fluid for supersonic case arises from the courant number restriction.

Table 3: Fluid Properties
Case m

$m$

g/mol

$g/mol$

cp

${c}_{p}$

J/kgK

$J/kgK$

mu

$mu$

N/ms

$N/ms$

Pr

$Pr$

Subsonic 28.9

$28.9$

1005

$1005$

1.79×105

$1.79×{10}^{-5}$

1

$1$

Supersonic 11640.3

$11640.3$

2.5

$2.5$

1.8×105

$1.8×{10}^{-5}$

1

$1$

Boundary Conditions:

Table 4.1: Boundary Conditions for Subsonic Case
Parameter Inlet (ABC) Outlet (GHI) Wall (AEHIFC) Wedges (AGHEBA + AGIFCA)
Velocity 7.58 ms1

Wedge

Temperature 300 K

k

$k$

0.862 m2/s2

ω

$\omega$

484.269 s1

αt

${\alpha }_{t}$

0

$0$

(Calculated)

0

$0$

(Calculated)

Wall Function Wedge
μt

${\mu }_{t}$

0

$0$

(Calculated)

0

$0$

(Calculated)

Wall Function Wedge
Table 4.2: Boundary Conditions for Supersonic Case
Parameter Inlet (ABC) Outlet (GHI) Wall (AEHIFC) Wedges (AGHEBA + AGIFCA)

Wedge
Pressure 1.2999 Nm2

Wave Transmissive 0.0296 Nm2

Temperature 1.0388 K

k

$k$

3.75×105 m2/s2

ω

$\omega$

0.144 s1

αt

${\alpha }_{t}$

0

$0$

(Calculated)

0

$0$

(Calculated)

Wall Function Wedge
μt

${\mu }_{t}$

0

$0$

(Calculated)

0

$0$

(Calculated)

Wall Function Wedge

## Results

Results for the subsonic cases are calculated from Bernoulli’s equation and the ideal gas equation as follows:

### Subsonic Flow

Subsonic flow does not see a significant temperature rise. So the speed of sound remains almost constant and can be calculated as:

c=γRTm−−−−−√

$c=\sqrt{\frac{\gamma RT}{m}}$

Here, γ

$\gamma$

T

$T$

m

$m$

and R

$R$

represent the specific heat ratio, temperature, and molecular weight of the fluid, and the universal gas constant respectively. The Mach number can then be calculated as:

M=AinUinAc

$M=\frac{{A}_{in}{U}_{in}}{Ac}$

Assuming a stagnation pressure of Pstag=0.1301 MPa

, the static pressure can be computed as:

Pstat=Pstag12ρu2

${P}_{stat}={P}_{stag}-\frac{1}{2}\rho {u}^{2}$

### Supersonic Flow

The relation between nozzle cross-section area A

$A$

and Mach number M

$M$

is what governs flow characteristics in supersonic flow through a de Laval nozzle [1]:

AAt=1M[2γ+1(1+γ12M2)]γ+12(γ1)

$\frac{A}{{A}_{t}}=\frac{1}{M}{\left[\frac{2}{\gamma +1}\left(1+\frac{\gamma -1}{2}{M}^{2}\right)\right]}^{\frac{\gamma +1}{2\left(\gamma -1\right)}}$

Using the known area ratio, the mach number variation is calculated by solving the above equation. The pressure can be calculated using [1]:

p=p0(1+γ12M2)γ1γ

$p={p}_{0}{\left(1+\frac{\gamma -1}{2}{M}^{2}\right)}^{\frac{\gamma }{1-\gamma }}$

A comparison of the Mach number and pressure variation in the nozzle obtained with SimScale with analytical results is given in Fig.3A and 3B for subsonic and Fig.4A and 4B for supersonic flow.

Fig.3. Visualization of Mach number and pressure (A, B) along the nozzle for subsonic flow

Fig.4. Visualization of Mach number and pressure (A, B) along the nozzle for supersonic flow

The deviation from analytical results exists because the latter is calculated with a one-dimensional hypothesis. Thus, all parameters are assumed to be uniform in the radial direction. Fig.5. shows that this is in fact not the case – there exists radial variation in flow variables. This is one reason why some deviation is seen between the two.

Fig.5. Contours of velocity and temperature in the nozzle. Clearly, there exists radial variation in these parameters.

## Disclaimer

This offering is not approved or endorsed by OpenCFD Limited, producer and distributor of the OpenFOAM software and owner of the OPENFOAM® and OpenCFD® trade marks. OPENFOAM® is a registered trade mark of OpenCFD Limited, producer and distributor of the OpenFOAM software.