#### Date of Award

2019

#### Document Type

Thesis

#### Degree Name

Master of Science in Engineering (MSE)

#### Department

Mechanical and Aerospace Engineering

#### Committee Chair

Sarma Rani

#### Committee Member

Sivaguru Ravindran

#### Committee Member

Kader Frendi

#### Subject(s)

Air ducts--Acoustic properties., Boundary value problems., Helmholtz equation., Neumann problem.

#### Abstract

Acoustic wave propagation in a duct is governed by the Helmholtz equation, which can be derived from the fluctuating forms of the mass, momentum, and energy balance equations (after considering harmonic dependence on time). In one-dimensional (1-D) domains, the Helmholtz equation is a second-order ordinary differential equation (ODE), while the fluctuating balance equations are all first-order ODEs. As a result, one needs two boundary conditions for the spatial pressure fluctuation $\hat{p}(x)$ in order to solve the Helmholtz equation, while only one boundary condition each is needed for $\hat{\rho}$, $\hat{u}$ and $\hat{p}$ in case of the fluctuating balance equations. Accordingly, this study was motivated by two principal objectives. The first was to develop the exact Neumann (or derivative) boundary condition at the inlet to a quasi 1-D duct needed to solve the Helmholtz equation. Such an exact boundary condition would ensure that the spatial pressure fluctuations $\hat{p}(x)$ obtained by solving the Helmholtz equation are identical to the $\hat{p}(x)$ obtained through the solution of the fluctuating balance equations. The second principal objective was to determine the exact relationship between the density and pressure fluctuations, $\hat{\rho}$ and $\hat{p}$ respectively, so that the $\hat{\rho}(x)$ calculated using the Helmholtz-equation $\hat{p}(x)$ is again identical to the $\hat{\rho}(x)$ obtained by solving the fluctuating balance equations. The exact $\hat{\rho}$-$\hat{p}$ relation is also compared with the ``classical" relation, namely $\hat{\rho} = \hat{p}/\bar{c}^2$, enabling us to evaluate the accuracy of the latter. The Neumann boundary conditions and the $\hat{\rho}$-$\hat{p}$ relations were developed for five cases with axially uniform and non-uniform duct cross-sectional areas, as well as homogeneous and inhomogeneous mean flow properties such as the velocity, temperature, density and pressure. It is seen that the $\hat{\rho} = \hat{p}/\bar{c}^2$ relation is valid only for the cases with uniform cross-sectional area and homogeneous mean properties. For the cases with uniform cross-section, inhomogeneous mean properties (with zero or uniform mean flow), the ``classical" $\hat{\rho}$ relation suffers from significant errors both in amplitude and phase.

#### Recommended Citation

Basu, Sattik, "On the Neumann boundary condition for the acoustic-wave Helmholtz equation, and the relationship between pressure and density fluctuations" (2019). *Theses*. 300.

https://louis.uah.edu/uah-theses/300