Application of Acoustic-Vortex Method for CFD-CAA Modelling of Multicopter Noise

Abstract

Quadcopters have become extremely popular and are used in areas ranging from monitoring traffic or fire conditions to distributing the Internet or cold drinks. Over the past ten years, there has been a sharp increase in the use of multicopters for various purposes. The noiselessness and efficiency of a propeller propulsion system are critical aspects in developing modern unmanned aerial vehicles. The development of this area of aviation technology in the context of tightening noise standards is impossible without effective optimization methods that work in conjunction with computer-aided design systems. Such a challenge requires the development of theoretical approaches to the numerical simulation of sound generation mechanisms by the propellers of multicopters and the corresponding software. This article discusses software based on a method for calculating sound generation and noise emission by a drone propeller, considering the decomposition of the vortex and acoustic modes. The development of this method makes it possible to consider the influence of a flow’s inhomogeneity and turbulence, rotor interference, sound diffraction by airframe elements, impedance characteristics of the hull coating, and other factors while ensuring the accuracy and speed of calculations.

APPENDIX

1.1 Acoustic-Vortex Decomposition of the Equations of Motion of a Compressible Medium

The acoustic and vortex modes of motion of the working fluid are introduced, since, based on the C-auchy–Helmholtz theorem, the velocity of the compressible fluid can be represented as a vector sum of the main translational and rotational motion of the fluid as an incompressible medium (the vortex mode) and small oscillations due to compressibility (the acoustic mode).

The following assumptions have been made:

—subsonic flow;

—isentropic flow;

—damping due to viscosity is not taken into account, since in the considered frequency range of the order of 1000–3000 Hz, the corresponding damping coefficient ([47], p. 18) in air is less than $\alpha = {{10}^{{ - 7}}}$ m^–1;

—acoustic oscillations (due to the compressibility of the medium) are significantly smaller than the vortex oscillations (the vortex and translational motion of the fluid).

The transfer of vortex disturbances, which arise as a result of the movement of a periodically inhomogeneous flow with the circumferential speed of the rotor, is considered as the main physical reason for the nonstationary process of generating sound by a multicopter propeller at the blade passing frequencies. The basic equations of motion of a compressible medium are written in the following form:

$${{\partial {\mathbf{V}}} \mathord{\left/ {\vphantom {{\partial {\mathbf{V}}} {\partial t}}} \right. \kern-0em} {\partial t}} + \nabla {{V}^{2}}{\text{/}}2 - {\mathbf{V}} \times \left( {\nabla \times {\mathbf{V}}} \right) = - \nabla P{\text{/}}\rho + \nu \Delta {\mathbf{V}},$$

(A.1)

$$\partial \rho {\text{/}}\partial t + \nabla \left( {\rho {\mathbf{V}}} \right) = 0,$$

(A.2)

$$s = \operatorname{const} .$$

(A.3)

In an isentropic flow, the increments of enthalpy, pressure, and density are related by the thermodynamic relations

$$di = dP{\text{/}}\rho ,~\quad dP = a_{0}^{2}d\rho ,$$

(A.4)

where ${{a}_{0}}$ is the speed of sound in the working environment. Taking into account relations (A.4), we rewrite Eqs. (A.1) and (A.2) in a form convenient for further transformations:

$$\frac{{\partial {\mathbf{V}}}}{{\partial t}} + \nabla \frac{{{{V}^{2}}}}{2} - {\mathbf{V}} \times (\nabla \times {\mathbf{V}}) = - \nabla i + \nu \Delta {\mathbf{V}},$$

(A.5)

$$\frac{1}{{a_{0}^{2}}}\left( {\frac{{\partial i}}{{\partial t}} + \left( {\nabla {\mathbf{V}}} \right)i} \right) + \nabla {\mathbf{V}} = 0.$$

(A.6)

Based on the Cauchy–Helmholtz theorem, the unsteady fluid velocity can be defined as the sum of the velocity U of the vortex mode (translational and rotational motion of an absolutely incompressible medium) and acoustic motion velocity ${{V}_{a}}$.

We introduce the scalar function: the acoustic potential φ. Then the acoustic velocity

$${{V}_{a}} = \nabla \varphi .$$

(A.7)

Thus, for the fluid velocity we obtain the following expression:

$${\mathbf{V}} = {\mathbf{U}} + \nabla \varphi = {\mathbf{U}} + {{V}_{a}}.$$

(A.8)

We will consider the subsonic flow $M = U{\text{/}}{{a}_{0}} \ll 1$ with small acoustic oscillations (${{V}_{a}} \ll {{a}_{0}}$). We also write down the following obvious relations:

$$\nabla {\mathbf{U}} = 0,\quad \nabla \times {\mathbf{V}} = \nabla \times {\mathbf{U}},$$

(A.9)

since

$$\nabla \times {{V}_{a}} \equiv \nabla \times \nabla \varphi \equiv 0.$$

(A.10)

Thus, the vortex disturbances of the flow are determined by the velocity of the incompressible flow. We now substitute relation (A.8) into Eq. (A.5). After simple transformations, from Eq. (A.5) we obtain

$$d{\mathbf{U}}{\text{/}}dt = - \nabla H + \nu \Delta {\mathbf{U}} + \nabla \varphi \times \nabla \times {\mathbf{U}},$$

(A.11)

where

$$H = i + d\varphi {\text{/}}dt + 0.5{{(\nabla \varphi )}^{2}},$$

(A.12)

$$d{\text{/}}dt = \partial {\text{/}}\partial t + {\mathbf{U}}\nabla .$$

(A.13)

In Eq. (A.11), the term $\nabla \varphi \times \nabla \times {\mathbf{U}}$ reflects the interaction of the acoustic and vortex modes. It can be important in the zone of acoustic resonance and the formation of feedback between acoustic waves and the mechanism of the emergence of instability waves in the mixing layer or in the formation of concentrated vortices, which, in turn, generate acoustic oscillations. In this case, this term can be neglected.

At the same time, (A.11) takes into account the action of viscous forces, which are involved in the formation of the inhomogeneity of the distribution of flow parameters over the pitch of the propeller blades and the source of the pressure oscillations at the BPFs.

Now, expressing i from formula (A.12) and substituting it into Eq. (A.6), while simultaneously dividing the velocity V in it into the acoustic and vortex modes, we obtain

$$\frac{1}{{a_{0}^{2}}}\frac{d}{{dt}}\left( {\frac{{d\varphi }}{{dt}} + \frac{{{{{(\nabla \varphi )}}^{2}}}}{2}} \right) - \Delta \varphi + \frac{1}{{a_{0}^{2}}}\nabla \varphi \nabla \left( {\frac{{d\varphi }}{{dt}} + \frac{{{{{(\nabla \varphi )}}^{2}}}}{2}} \right) = \frac{1}{{a_{0}^{2}}}\left( {\frac{{dH}}{{dt}} + \nabla \varphi \nabla H} \right).$$

(A.14)

Taking into account the previously adopted assumptions, we can linearize Eq. (A.14) with respect to φ, and it will take the form

$$\frac{1}{{a_{0}^{2}}}\frac{{{{d}^{2}}\varphi }}{{d{{t}^{2}}}} - \Delta \varphi = \frac{1}{{a_{0}^{2}}}\frac{{dH}}{{dt}}.$$

(A.15)

Equations (A.14) and (A.15) show that the unsteady vortex motion of the fluid generates acoustic oscillations. The term $\frac{1}{{a_{0}^{2}}}\frac{{dH}}{{dt}}$ in the right-hand side of the equation acts as the source of acoustic oscillations in the inhomogeneous wave acoustic equation.

At the same time, the energy of the acoustic mode can be partially transformed into the energy of the vortex motion, which is expressed by the term $\nabla \varphi \times \nabla \times {\mathbf{U}}$ in Eq. (A.11). In this model, the member $\nabla \varphi \times \nabla \times {\mathbf{U}}$ is omitted, since such an interaction is important in the zone of acoustic-vortex resonance.

Taking into account the last assumption, linearizing with respect to φ, and neglecting the convection of the acoustic mode, relations (A.11) and (A.12) are written in the following form:

$${{d{\mathbf{U}}} \mathord{\left/ {\vphantom {{d{\mathbf{U}}} {dt}}} \right. \kern-0em} {dt}} = - \nabla H + \nu \Delta {\mathbf{U}},$$

(A.16)

$$ - {{\partial \varphi } \mathord{\left/ {\vphantom {{\partial \varphi } {\partial t}}} \right. \kern-0em} {\partial t}} = i - H.$$

(A.17)

Thus, Eqs. (A.16) and (A.17) provide a solution to the problem of splitting the basic equations of motion of a compressible fluid into the vortex and acoustic modes. Equation (A.16) describes the vortex turbulent motion of an incompressible viscous fluid under the action of the unsteady pressure gradient $\nabla {{P}_{{v}}} = {{\rho }_{0}}\nabla H$.

We differentiate Eq. (A.15) by taking the partial derivative with respect to time and substituting into it the expression for $\partial \varphi {\text{/}}\partial t$ from formula (A.17). After simple transformations, we get

$$\frac{1}{{{{a}^{2}}}}\frac{{{{d}^{2}}i}}{{d{{t}^{2}}}} - \Delta i = - \Delta H.$$

(A.18)

The disturbing function on the right side of Eq. (A.18) is calculated from the velocity field of the vortex mode after solving Eq. (A.16)

$$ - \Delta H = \nabla \cdot \left( {{\mathbf{U}} \cdot \nabla {\mathbf{U}}} \right) = \nabla \cdot \left( {\nabla \left( {\frac{1}{2}{{U}^{2}}} \right) - {\mathbf{U}} \times \nabla \times {\mathbf{U}}} \right).$$

(A.19)

Neglecting the convective terms in the time derivative of Eq. (A.18), we obtain

$$\frac{1}{{{{a}^{2}}}}\frac{{{{\partial }^{2}}i}}{{\partial {{t}^{2}}}} - \Delta i = S,$$

(A.20)

where S denotes the disturbing function, which is determined from the field of incompressible flow velocities

$$S = \nabla \left( {\nabla \left( {\frac{1}{2}{{U}^{2}}} \right) - {\mathbf{U}} \times \nabla \times {\mathbf{U}}} \right).$$

(A.21)

For an undisturbed flow, the acoustic potential $\varphi = 0$ and

$${{i}_{0}} = {{H}_{0}};\quad {{S}_{0}} = - {{H}_{0}}.$$

(A.22)

The pulsating part of the function S is denoted by s. Functions $i$, H, and s can be expressed in terms of the average values and pulsation components

$$h = i - {{i}_{0}};\quad g = H - {{H}_{0}};\quad s = S - {{S}_{0}}.$$

(A.23)

The amplitude of the pressure pulsations is two to three orders of magnitude lower than the average pressure; thus, for the enthalpy oscillations, we can approximately write

$$h \approx {{(P - {{P}_{0}})} \mathord{\left/ {\vphantom {{(P - {{P}_{0}})} {{{\rho }_{0}}}}} \right. \kern-0em} {{{\rho }_{0}}}} = {{P{\kern 1pt} '} \mathord{\left/ {\vphantom {{P{\kern 1pt} '} {{{\rho }_{0}}}}} \right. \kern-0em} {{{\rho }_{0}}}}.$$

(A.24)

Similarly for oscillations of the vortex mode (pseudosound)

$$g \approx {{(H - {{H}_{0}})} \mathord{\left/ {\vphantom {{(H - {{H}_{0}})} {{{\rho }_{0}}}}} \right. \kern-0em} {{{\rho }_{0}}}} = P_{{v}}^{'}{\text{/}}{{\rho }_{0}},$$

(A.25)

where $P{\kern 1pt} '$ is the pressure of sound and $P_{{v}}^{'}$ is the pseudosound. Taking (A.17) into account,

$$P{\kern 1pt} ' = P_{{v}}^{'} - {{\rho }_{0}}{{\partial \varphi } \mathord{\left/ {\vphantom {{\partial \varphi } {\partial t}}} \right. \kern-0em} {\partial t}}.$$

(A.26)

The last expression clearly shows that the pressure pulsations are equal to the sum of the pulsations from the nonstationary vortex motion (as an incompressible medium): the pseudosound and acoustic oscillations.

We write the acoustic-vortex equation in the Cartesian coordinate system

$$\frac{1}{{{{a}^{2}}}}\frac{{{{\partial }^{2}}h}}{{\partial {{t}^{2}}}} - \frac{{{{\partial }^{2}}h}}{{\partial {{x}^{2}}}} - \frac{{{{\partial }^{2}}h}}{{\partial {{y}^{2}}}} - \frac{{{{\partial }^{2}}h}}{{\partial {{z}^{2}}}} = s.$$

(A.27)

This wave equation is solved by an explicit method in time for each given harmonic of the propeller’s BPF (the BPF is equal to the frequency of the rotor’s rotation in hertz multiplied by the number of blades). The disturbing function on the right side is determined by the harmonic analysis of the function (A.21) taking into account (A.23) before the solution of the wave equation. Thus, in Eq. (A.27) and further in the text, the harmonic number of the blade repetition rate for h and s is omitted. As a result of the solution, the amplitudes of the BPF harmonics at each point in space can be obtained.

The finite-difference analogs of the differential equations in the Cartesian coordinate system are obtained by integrating the acoustic-wave equation over space and time with the introduction of finite volumes.

The entire flow field is covered with a rectangular grid. Each grid node is assigned three numbers (i, j, k), which determine the ordinal number of the final volume (cell) on the x, y, and z coordinate axes. The boundaries between neighboring cells pass through the middle of the grid steps.

In addition, we introduce a time grid with superscript (m) and the uniform time step $\Delta t$, in which each moment of time corresponds to number m so that

$$t + \Delta t = (m + 1)\Delta t.$$

(A.28)

We consider the derivation of the finite difference equations for the internal grid nodes (cell 1 in Fig. 13). For Eq. (A.27) we use the integral method: we integrate Eq. (A.27) over space and time within one cell and one time step

$$\mathop \smallint \limits_{t - \Delta t/2}^{t + \Delta t/2} \,\mathop \smallint \limits_{{{x}_{1}}}^{{{x}_{2}}} \,\mathop \smallint \limits_{{{y}_{1}}}^{{{y}_{2}}} \,\mathop \smallint \limits_{{{z}_{1}}}^{{{z}_{2}}} \left( {\frac{1}{{{{a}^{2}}}}\frac{{{{\partial }^{2}}h}}{{\partial {{t}^{2}}}} - \frac{{{{\partial }^{2}}h}}{{\partial {{x}^{2}}}} - \frac{{{{\partial }^{2}}h}}{{\partial {{y}^{2}}}} - \frac{{{{\partial }^{2}}h}}{{\partial {{z}^{2}}}} - s} \right)\partial t\partial x\partial y\partial z = 0.$$

(A.29)

Fig. 13.

Types of calculation cells: (1) internal, (2) wall, (3) impedance boundary.

Here, for the inner cell 1, the limits of integration over volume

$$\begin{gathered} {{x}_{1}} = x - \frac{1}{2}\Delta x{\text{,}}\quad {{y}_{1}} = y - \frac{1}{2}\Delta y{\text{,}}\quad {{z}_{1}} = z - \frac{1}{2}\Delta z, \\ {{x}_{2}} = x + \frac{1}{2}\Delta x{\text{,}}\quad {{y}_{2}} = y + \frac{1}{2}\Delta y,\quad {{z}_{2}} = z - \frac{1}{2}\Delta z. \\ \end{gathered} $$

(A.30)

We assume that the pressure, its second time derivative, and the distrubing function are constant inside the cell volume. Then we can write for each harmonic of the BPF

$$\begin{gathered} \frac{1}{{{{a}^{2}}}}\int\limits_{t - \Delta t/2}^{t + \Delta t/2} {\frac{{{{\partial }^{2}}h}}{{\partial {{t}^{2}}}}} \partial t\int\limits_{{{x}_{1}}}^{{{x}_{2}}} {dx} \int\limits_{{{y}_{1}}}^{{{y}_{2}}} {dy} \int\limits_{{{z}_{1}}}^{{{z}_{2}}} {dz} \\ - \;\int\limits_{t - \Delta t/2}^{t + \Delta t/2} {\left( {\int\limits_{{{x}_{1}}}^{{{x}_{2}}} {\int\limits_{{{y}_{1}}}^{{{y}_{2}}} {\int\limits_{{{z}_{1}}}^{{{z}_{2}}} {\left( {\frac{{{{\partial }^{2}}h}}{{\partial {{x}^{2}}}} - \frac{{{{\partial }^{2}}h}}{{\partial {{y}^{2}}}} - \frac{{{{\partial }^{2}}h}}{{\partial {{z}^{2}}}}} \right)dxdydz} } } } \right)} {\kern 1pt} {\kern 1pt} dt - \int\limits_{t - \Delta t/2}^{t + \Delta t/2} s \int\limits_{{{x}_{1}}}^{{{x}_{2}}} {dx} \int\limits_{{{y}_{1}}}^{{{y}_{2}}} {dy} \int\limits_{{{z}_{1}}}^{{{z}_{2}}} {dz} = 0. \\ \end{gathered} $$

(A.31)

Taking into account formulas for the finite-difference analogs of the derivatives

$$\int\limits_{t - \Delta t/2}^{t + \Delta t/2} {\frac{{{{\partial }^{2}}h}}{{\partial t}}dt} = \left. {\frac{{\partial h}}{{\partial t}}} \right|_{{t - \Delta t/2}}^{{t + \Delta t/2}} = \frac{{h_{{ijk}}^{{m + 1}} - h_{{ijk}}^{m}}}{{\Delta t}} - \frac{{h_{{ijk}}^{m} - h_{{ijk}}^{{m - 1}}}}{{\Delta t}} = \frac{{h_{{ijk}}^{{m + 1}} - 2h_{{ijk}}^{m} + h_{{ijk}}^{{m - 1}}}}{{\Delta t}}$$

(A.32)

and Eq. (A.31) is transformed to the form

$$h_{{ijk}}^{{m + 1}} = 2h_{{ijk}}^{m} - h_{{ijk}}^{{m - 1}} + \frac{{\Delta t}}{{{{a}^{2}}W}}{{\Psi }_{c}}.$$

(A.33)

Here, the harmonic number of the BPF is omitted and $W$ is the volume of the cell. Function ${{\Psi }_{c}}$ (c is the cell type) is determined by the expression

$$\begin{gathered} {{\Psi }_{c}} = \int\limits_{t - \Delta t/2}^{t + \Delta t/2} {\left[ {\int\limits_{{{y}_{1}}}^{{{y}_{2}}} {\int\limits_{{{z}_{1}}}^{{{z}_{2}}} {\left( {{{{\left. {\frac{{{{\partial }^{2}}h}}{{\partial {{x}^{2}}}}} \right|}}^{{{{x}_{2}}}}} - {{{\left. {\frac{{{{\partial }^{2}}h}}{{\partial {{x}^{2}}}}} \right|}}_{{{{x}_{1}}}}}} \right)} } dydz} \right]dt} + \int\limits_{t - \Delta t/2}^{t + \Delta t/2} {\left[ {\int\limits_{{{z}_{1}}}^{{{z}_{2}}} {\int\limits_{{{x}_{1}}}^{{{x}_{2}}} {\left( {{{{\left. {\frac{{{{\partial }^{2}}h}}{{\partial {{y}^{2}}}}} \right|}}^{{{{y}_{2}}}}} - {{{\left. {\frac{{{{\partial }^{2}}h}}{{\partial {{y}^{2}}}}} \right|}}_{{{{y}_{1}}}}}} \right)} } dzdx} \right]dt} \\ {\text{ + }}\;\int\limits_{t - \Delta t/2}^{t + \Delta t/2} {\left[ {\int\limits_{{{y}_{1}}}^{{{y}_{2}}} {\int\limits_{{{x}_{1}}}^{{{x}_{2}}} {\left( {{{{\left. {\frac{{{{\partial }^{2}}h}}{{\partial {{z}^{2}}}}} \right|}}^{{{{z}_{2}}}}} - {{{\left. {\frac{{{{\partial }^{2}}h}}{{\partial {{z}^{2}}}}} \right|}}_{{{{z}_{1}}}}}} \right)} } dydx} \right]dt} + \int\limits_{t - \Delta t/2}^{t + \Delta t/2} {s\partial t} \int\limits_{{{x}_{1}}}^{{{x}_{2}}} {dx} \int\limits_{{{y}_{1}}}^{{{y}_{2}}} {dy} \int\limits_{{{z}_{1}}}^{{{z}_{2}}} {dz} = 0 \\ \end{gathered} $$

(A.34)

and depends on the cell type. Thus, for inner cells such as 1 (see Fig. 13)

$${{\left. {\frac{{\partial h}}{{\partial x}}} \right|}^{{{{x}_{1}}}}} = \frac{{h_{{i + 1,j,k}}^{m} - h_{{i,j,k}}^{m}}}{{\Delta x}},\quad {{\left. {\frac{{\partial h}}{{\partial x}}} \right|}_{{{{x}_{2}}}}} = \frac{{h_{{i,j,k}}^{m} - h_{{i - 1,j,k}}^{m}}}{{\Delta x}},$$

(A.35)

for cells located on the wall normal to the x axis on the right, we get

$${{\left. {\frac{{\partial h}}{{\partial x}}} \right|}^{{{{x}_{1}}}}} = 0,\quad {{\left. {\frac{{\partial h}}{{\partial x}}} \right|}_{{{{x}_{2}}}}} = \frac{{h_{{i,j,k}}^{m} - h_{{i - 1,j,k}}^{m}}}{{\Delta x}},$$

(A.36)

and for cells located on the impedance boundary with the normal n, and also, if it is necessary to take into account the pseudosonic oscillations

$${{\left. {\frac{{\partial h}}{{\partial n}}} \right|}^{n}} = - \frac{1}{{aZ}}\left( {\frac{{{{h}^{{m + 1}}} - {{h}^{{m - 1}}}}}{{2\Delta t}}} \right) + \frac{{\partial g}}{{\partial n}} + \frac{1}{{aZ}}\frac{{\partial g}}{{\partial t}}.$$

(A.37)

Thus, the corresponding derivative of h along the normal to the boundary surface is determined through the time derivative and the specific complex acoustic impedance Z on the given boundary for the corresponding BPF harmonic. The derivatives of the oscillations of the vortex mode are known from the solution for the unsteady flow of an incompressible viscous medium.