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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 \(k-\omega\ SST\) model was used to model turbulence. Simulaton 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.).

CompressibleFlow-geometry

Fig.1. Geometry of the de Laval Nozzle

Table 1: Point Coordinates
Point \(x\) \(y\) \(z\)
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 \(y^+ = 30\) for the subsonic and \(300\) for the supersonic case. In order to keep the flow two-dimensional, the mesh was designed to have only one layer in the \(y\) 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
CompressibleFlow-mesh

Fig.2. Mesh used for the SimScale case

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 \(m\) \(g/mol\) \(c_p\) \(J/kgK\) \(mu\) \(N/ms\) \(Pr\)
Subsonic \(28.9\) \(1005\) \(1.79\times 10^{-5}\) \(1\)
         
Supersonic \(11640.3\) \(2.5\) \(1.8\times 10^{-5}\) \(1\)

Boundary Conditions:

Table 4.1: Boundary Conditions for Subsonic Case
Parameter Inlet (ABC) Outlet (GHI) Wall (AEHIFC) Wedges (AGHEBA + AGIFCA)
Velocity \(7.58\ ms^{-1}\) Zero Gradient \(0.0\ ms^{-1}\) Wedge
Pressure Zero Gradient \(1.3\times 10^5\ Nm^{-2}\) Zero Gradient Wedge
Temperature \(300\ K\) Zero Gradient Zero Gradient Wedge
\(k\) \(0.862\ m^2/s^2\) Zero Gradient Wall Function Wedge
\(\omega\) \(484.269\ s^{-1}\) Zero Gradient Wall Function Wedge
\(\alpha _t\) \(0\) (Calculated) \(0\) (Calculated) Wall Function Wedge
\(\mu _t\) \(0\) (Calculated) \(0\) (Calculated) Wall Function Wedge
Table 4.2: Boundary Conditions for Supersonic Case
Parameter Inlet (ABC) Outlet (GHI) Wall (AEHIFC) Wedges (AGHEBA + AGIFCA)
Velocity Zero Gradient Zero Gradient \(0.0\ ms^{-1}\) Wedge
Pressure \(1.2999\ Nm^{-2}\) Wave Transmissive \(0.0296\ Nm^{-2}\) Zero Gradient Wedge
Temperature \(1.0388\ K\) Zero Gradient Zero Gradient Wedge
\(k\) \(3.75\times 10^{-5}\ m^2/s^2\) Zero Gradient Wall Function Wedge
\(\omega\) \(0.144\ s^{-1}\) Zero Gradient Wall Function Wedge
\(\alpha _t\) \(0\) (Calculated) \(0\) (Calculated) Wall Function Wedge
\(\mu _t\) \(0\) (Calculated) \(0\) (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:

\[c = \sqrt{\frac{\gamma RT}{m}}\]

Here, \(\gamma\), \(T\), \(m\) and \(R\) 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:

\[M = \frac{A_{in}U_{in}}{Ac}\]

Assuming a stagnation pressure of \(P_{stag} = 0.1301\ MPa\), the static pressure can be computed as:

\[P_{stat} = P_{stag} - \frac{1}{2}\rho u^2\]

Supersonic Flow

The relation between nozzle cross-section area \(A\) and Mach number \(M\) is what governs flow characteristics in supersonic flow through a de Laval nozzle [1]:

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

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

\[p = p_0\left(1+\frac{\gamma -1}{2}M^2\right)^{\frac{\gamma}{1-\gamma}}\]

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.

MSub PSub

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

MSup PSup

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.

ContU ContT

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.