Acoustic Modal Analysis

The Acoustic Modal Analysis model computes the acoustic frequencies, normalized mode shapes and their linear growth rates for a given geometry and CFD solution.

Acoustic Wave Equation

The acoustic wave equation (the inhomogeneous Helmholtz equation) describes sound waves in a fluid:

Figure 1. EQUATION_DISPLAY

\frac{\partial^{2} p^{'}}{\partial t^{2}} - \nabla\cdot ({\bar{c}}^{2} \nabla p^{'}) = (γ - 1) \frac{\partial {\dot{q}}^{'}}{\partial t}

(4770)

where

p^{'}

is the acoustic pressure fluctuation,

t

is the time,

\bar{c}

is the mean speed of sound,

γ

is the specific heat ratio, and

{\dot{q}}^{'}

is the heat release fluctuation.

Harmonic Fluctuations

Assuming harmonic fluctuations at frequency

f = \frac{ω}{2 π}

(where

ω

is a complex number), the pressure fluctuation

p^{'}

and the heat release fluctuation

{\dot{q}}^{'}

are expressed as:

Figure 2. EQUATION_DISPLAY

p^{'} = \hat{p} e^{(- i ω t)}

(4771)

and

Figure 3. EQUATION_DISPLAY

{\dot{q}}^{'} = \hat{q} e^{(- i ω t)}

(4772)

respectively.

where:

$\hat{p}$ is the pressure fluctuation magnitude,
$\hat{q}$ is the heat release fluctuation magnitude.

Substituting Eqn. (4771) and Eqn. (4772) into the acoustic wave equation Eqn. (4770) gives:

Figure 4. EQUATION_DISPLAY

ω^{2} \hat{p} + \nabla\cdot ({\bar{c}}^{2} \nabla \hat{p}) = i ω (γ - 1) \hat{q}

(4773)

Assuming that

p^{'}

and

{\dot{q}}^{'}

denote the cell center values, integrating Eqn. (4773) over the cell volume gives:

Figure 5. EQUATION_DISPLAY

\int ω^{2} \hat{p} d V + \int \nabla\cdot ({\bar{c}}^{2} \nabla \hat{p}) d V = \int i ω (γ - 1) \hat{q} d V

(4774)

Discretizing Eqn. (4774) gives the following matrix eigenvalue equation—whose eigenvalues are the acoustic frequencies

ω

, and the corresponding eigenvectors are the acoustic mode shapes

\hat{P}

Figure 8. EQUATION_DISPLAY

[A + ω B (ω) + ω^{2} C] \hat{P} = D (ω) \hat{P}

(4777)

where matrix

A

accounts for the wave propagation term,

B (ω)

accounts for the impedance boundary conditions,

C

is equal to the identity matrix

I

(except when using the quadratic impedance boundary), and

D (ω)

denotes the combustion heat release source term in the Flame Transfer Function (FTF)—which is omitted from Eqn. (4777) for non-reactive flows. (See Flame Transfer Function).

\hat{P}

is a size

N_{c e l l s}

vector for the acoustic mode (eigenvector) and

ω

is the acoustic frequency (eigenvalue).

$A$ , $B$ , $C$ , and $D$ are sparse matrices of size $N_{c e l l s} \times N_{c e l l s}$ in which $N_{c e l l s}$ is the number of cells that are in the mesh. When discretizing the diffusion term in Eqn. (4774), Simcenter STAR-CCM+ provides the option of including the secondary gradient term, or omitting it.

The phase of the acoustic pressure

\hat{p}

is calculated as:

Figure 14. EQUATION_DISPLAY

φ = \arctan (\frac{I m (\hat{p})}{Re (\hat{p})})

(4783)

Unstable Modes

The acoustic frequency is composed of real (

ω_{r}

) and imaginary (

ω_{i}

) parts:

Figure 15. EQUATION_DISPLAY

ω = ω_{r} + i ω_{i}

(4784)

Using the above expression, the pressure fluctuation can be expressed as:

p^{'} = \hat{p} e^{- i ω t} = [\hat{p} e^{ω_{i} t}] e^{- i ω_{r} t}

(4785)

When

ω_{i} < 0

, the pressure fluctuations decay with time—and the mode is stable. If

ω_{i} > 0

, the pressure fluctuations grow with time—and the mode is unstable. In situations where

ω_{i} = 0

, the amplitude of the pressure fluctuations remains constant.

Boundary Conditions

Simcenter STAR-CCM+ allows you to define the acoustic boundary condition in the following ways:

Perfectly Reflecting (Hard Wall) Boundary: Figure 16. EQUATION_DISPLAY

$\nabla \hat{p} \cdot n_{B C} = 0$

(4786)
Zero Acoustic Pressure Boundary: Figure 17. EQUATION_DISPLAY

$\hat{p} = 0$

(4787)
Specified (Constant) Impedance Boundary: Figure 18. EQUATION_DISPLAY

$\bar{c} Z \nabla \hat{p} \cdot n_{B C} - i ω \hat{p} = 0$

(4788)
where the acoustic impedance specifies the magnitude and phase of the reflected wave, and is the same for every frequency.
Quadratic Impedance Profile Boundary: The acoustic impedance is a function of acoustic frequency $Z \equiv Z (ω)$ and the impedance is specified using the quadratic form:
Figure 19. EQUATION_DISPLAY

$\frac{1}{Z} = \frac{1}{Z_{0}} + ω Z_{1} + \frac{Z_{2}}{ω}$

(4789)
in which $Z_{0}$ , $Z_{1}$ , and $Z_{2}$ are complex valued constants.