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.

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.).

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

Table 1: Point Coordinates

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 :

Mesh type

Cells in x

Cells iny

Cells in z

Number of nodes

Type

blockMesh

150

1

100

15000

2D Hex

Table 2: Mesh Metrics

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.

Case

\(m\) [\(g/mol\)]

\(c_p\) [\(J/kgK\)]

\(\mu\) [\(N/ms\)]

\(Pr\)

Subsonic

28.9

1005

\(1.79\times10^{-5}\)

1

Supersonic

11640.3

2.5

\(1.8\times10^{-5}\)

1

Table 3: Fluid Properties

Boundary Conditions:

Parameter

Inlet (ABC)

Outlet (GHI)

Wall (AEHIFC)

Wedges (AGHEBA + AGIFCA)

Velocity

\(7.58\) \(m/s\)

Zero Gradient

\(0.0\) \(m/s\)

Wedge

Pressure

Zero Gradient

\(1.3\times10^{5}\) \(N/m^2\)

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 4.1: Boundary Conditions for Subsonic Case

Parameter

Inlet (ABC)

Outlet (GHI)

Wall (AEHIFC)

Wedges (AGHEBA + AGIFCA)

Velocity

Zero Gradient

Zero Gradient

\(0.0\) \(m/s\)

Wedge

Pressure

\(1.2999\) \N/m^2\)

Wave Transmissive \(0.0296\) \(N/m^2\)

Zero Gradient

Wedge

Temperature

\(1.0388\) \(K\)

Zero Gradient

Zero Gradient

Wedge

\(k\)

\3.75\times10^{-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

Table 4.2: Boundary Conditions for Supersonic Case

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 ^{2}:

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.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.

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