Determine the Expressions for the Derivatives Used in the Continuity Equation

Equation of Continuity

The continuity equations of both the photogenerated electrons and holes are solved to get the spatial distribution of these carriers and their concentrations are used to calculate the photocurrent density.

From: Photodetectors , 2016

Filtration Mechanisms and Theory

Irwin M. Hutten , in Handbook of Nonwoven Filter Media, 2007

2.2.2 The equations of motion and continuity

The equations of motion and continuity are the fundamental equations from which filtration theory is derived. The simplified and abbreviated explanations below are based on Bird, Lightfoot, and Stewart ⁽²⁸⁾ . Chapter 3 of this classic textbook contains a comprehensive teaching of the two equations for isothermal systems including the equation of mechanical energy.

2.2.2.1 The equation of continuity

The equation of continuity is simply a mass balance of a fluid flowing through a stationary volume element. It states that the rate of mass accumulation in this volume element equals the rate of mass in minus the rate of mass out. In vector form the balance is as follows:

(2.1) $\frac{\partial \rho }{\partial t}=-(\text{▾}\cdot \rho v)$

ρ is the density, kg/m³;

t is the time variable, s;

ν is the velocity vector, m/s;

(∇ ρν) is the vector operator indicating the divergence of the mass flux ρν.

Note that ∇ has the units of reciprocal length, m⁻¹.

2.2.2.2 The equation of motion

In the case of an incompressible fluid, Equation (2.1) simplifies to the equation of motion.

(2.2) $(\nabla ·\upsilon )=0$

Analogous to the equation of continuity, the equation of motion is a momentum balance around a unit volume of fluid. It states that rate of momentum accumulation equals the rate of momentum in minus the rate of momentum out plus the sum of all the other forces acting on the system. Its vector form is:

(2.3) $\rho \frac{D\upsilon }{Dt}=-\nabla p-[\nabla ·\tau ]+\rho g$

ρ is the density, kg/m³;

t is the time variable, s;

$\rho \frac{D\upsilon }{Dt}$ is the rate of momentum accumulation per unit volume, kg/m²/s²;

∇ _p is the pressure force on the element per unit volume, Pa (Pa = kg/m/s¹²);

[∇ τ] is the viscous force on the element per unit volume, kg/m²/s²;

τ is the shear stress tensor, kg/m/s²;

ρg is the gravitational force on the element per unit volume, kg/m²/s²; g is the acceleration of gravity (9.807 m/s²);

[∇ τ] can be revised in terms of the fluid viscosity μ, assuming constant μ and constant ρ.

(2.4) $[\nabla ·\tau ]=\mu {\nabla }^2\upsilon$

For the case of constant μ and constant ρ, Equation (2.3) becomes:

(2.5) $\rho \frac{D\upsilon }{Dt}=-\nabla p-[\mu {\nabla }^2\cdot v]+\rho g$

Equation (2.5) is known as the Navier–Stokes equation. For the case of [∇ τ] = 0, Equation (2.5) reduces to:

(2.6) $\rho \frac{D\upsilon }{Dt}=-\nabla p+\rho g$

Equation (2.6) is known as the Euler equation.

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MICROSCOPIC BALANCES

W. Fred Ramirez , in Computational Methods in Process Simulation (Second Edition), 1997

6.1 CONSERVATION OF TOTAL MASS (EQUATION OF CONTINUITY)

The equation of continuity is derived by writing a mass balance over an arbitrary finite volume element Δ xΔyΔz through which a fluid is flowing, as is shown in Figure 6.1. We then take the limit as the volume element goes to zero to get the appropriate equation for any point in the spatial domain.

Figure 6.1. Differential Element for Equation of Continuity.

The rate of mass entering the volume element perpendicular to the x axis at x is ρυ_x ΔyΔz | _x and at the plane x + Δx is ρυ_x ΔyΔz | _x+Δx Similar expressions can be written for the other planes. The rate of accumulation of mass within the volume element is ϑ(ρΔxΔyΔz)/ϑt. The mass balance therefore becomes

$\begin{array}{ccccc}\text{Rate of Accumulation of Mass}& =& \text{Rate of Mass In}& -& \text{Rate of Mass Out}\end{array}$

(6.1.1) $\begin{array}{cc}\frac{\partial \left(\rho \Delta x\Delta y\Delta z\right)}{\partial t}=& \Delta y\Delta z\left(\rho {\upsilon }_x|{}_x-\rho {\upsilon }_x|{}_{x+\Delta x}\right)\\ & +\Delta x\Delta z\left(\rho {\upsilon }_y|{}_y-\rho {\upsilon }_y|{}_{y+\Delta y}\right)\\ & +\Delta x\Delta y\left(\rho {\upsilon }_z|{}_z-\rho {\upsilon }_z|{}_{z+\Delta z}\right)\end{array}$

By dividing this equation by the volume element xΔyΔzΔ and taking the limit as the volume (ΔxΔyΔz) approaches zero, we get the point description

(6.1.2) $\frac{\partial \rho }{\partial t}=-\left(\frac{\partial }{\partial x}\rho {\upsilon }_x+\frac{\partial }{\partial y}\rho {\upsilon }_y+\frac{\partial }{\partial z}\rho {\upsilon }_z\right)$

This differential overall mass balance, which holds at any point within the volume of the system, is often called the equation of continuity. It describes the rate of change of the density of a fluid at any point in the system. The equation expressed in vector form is

The equation of continuity can be rearranged to become

(6.1.4) $\frac{\partial \rho }{\partial t}+{\upsilon }_x\frac{\partial \rho }{\partial x}+{\upsilon }_y\frac{\partial \rho }{\partial y}+{\upsilon }_z\frac{\partial \rho }{\partial z}=-\rho \left(\frac{\partial {\upsilon }_x}{\partial x}+\frac{\partial {\upsilon }_y}{\partial y}+\frac{\partial {\upsilon }_z}{\partial z}\right)$

The left side of equation (6.1.4) is the substantial derivative of the density, that is, the time derivative for a path following the fluid motion; and it can be written as

(6.1.5) $\begin{array}{c}\frac{D\rho }{Dt}=\frac{\partial \rho }{\partial t}+{\upsilon }_x\frac{\partial \rho }{\partial x}+{\upsilon }_y\frac{\partial \rho }{\partial y}+{\upsilon }_z\frac{\partial \rho }{\partial z}\end{array}$

A very important special form of the equation of continuity is that for a fluid of constant density (incompressible fluid), for which the substantial derivative is zero. Equation (6.1.4) therefore becomes for an incompressible fluid,

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Aortic dissection: Numerical modeling and virtual surgery

Igor B. Saveljic , Nenad Filipovic , in Computational Modeling in Bioengineering and Bioinformatics, 2020

6.1 Continuity equation

The equation of continuity is an analytic form of the law on the maintenance of mass. This law can be applied both to the elemental mass of the fluid particle dm and to the final mass m. According to this law, the mass of the fluid particle does not change during movement in an uninterrupted electric field. By applying Reynolds theorem to the law on the maintenance of mass, it is obtained as (Filipovic, 1999)

(5.1) $\frac{\mathit{Dm}}{\mathit{Dt}}=\frac{D}{\mathit{Dt}}\underset{V}{\int }\mathit{\rho dV}=\underset{V}{\int }\left(\frac{\mathit{D\rho }}{\mathit{Dt}}+\rho \frac{\partial v_i}{\partial x_i}\right)\mathit{dV}=0$

For an arbitrary control volume V, the equation of continuity in the point is obtained as

(5.2) $\frac{\mathit{D\rho }}{\mathit{Dt}}+\rho \frac{\partial v_i}{\partial x_i}=0$

After this, Reynolds theorem is reduced to the following form:

(5.3) $\frac{D}{\mathit{Dt}}\underset{V}{\int }\left(f\rho \right)\mathit{dV}=\underset{V}{\int }\left(\frac{\partial f}{\partial t}+v_i\frac{\partial f}{\partial x_i}\right)\mathit{\rho dV}=\underset{V}{\int }\rho \frac{\mathit{Df}}{\mathit{Dt}}\mathit{dV}$

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Fundamentals of Fluid Simulation by the MPS Method

Seiichi Koshizuka , ... Takuya Matsunaga , in Moving Particle Semi-implicit Method, 2018

2.2.2.2 Equation of Continuity

The equation of continuity is expressed as follows:

(2.7) $\frac{D\rho }{Dt}+\rho \nabla \cdot \mathbf{u}=0$

The equation of continuity expresses the law of mass conservation. Specifically, the equation of continuity expresses how fluid density changes according to the mass flow from a certain unit volume. The first term on the left-hand side of Eq. (2.7) expresses the time change of fluid density per unit domain. The second term on the left-hand side is the multiplication of the fluid density by the divergence of velocity. The divergence of velocity expresses the fluid volume which flows out from a unit volume per unit time. Therefore, the second term expresses the fluid mass that flows out from the unit volume per unit time because the fluid density is multiplied by the volume. (Although it appears necessary to divide the mass by the volume to calculate the density, the division is not necessary because the equation is considered per unit volume, and has already been divided by the volume.) We can make Eq. (2.7) easy to understand by transforming it as follows:

(2.8) $\frac{D\rho }{Dt}=-\rho \nabla \cdot \mathbf{u}$

From the above equation, we can easily understand that the fluid density decreases if the fluid mass flows out from the unit volume because $\nabla \cdot \mathbf{u}$ is positive, and the material derivative of fluid density becomes negative because coefficient $-\rho$ is multiplied by $\nabla \cdot \mathbf{u}$ . The negative sign is necessary for the right-hand side because the fluid density increases in the case where fluid flows in, while it decreases when fluid flows out. If Eq. (2.6) is not satisfied completely, some part of the fluid mass is lost, or the fluid mass is increased. Therefore, it is very important to satisfy the equation of continuity in grid-based methods. On the other hand, the MPS method can satisfy mass conservation very easily because the MPS method assumes every particle has a constant mass. If we do not generate or delete particles, the local and total fluid masses are completely conserved. This is an advantage of the MPS method. In the MPS method, the equation of continuity is indirectly considered in the calculation of pressure Poisson equation. The details are explained in Section 2.2.5. For the more detailed contents of hydrodynamics, see the references Hughes and Brighton (1999) and Potter and Wiggert (2007).

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Theoretical Background: an Outline of Computational Electromagnetics (CEM)

Dragan Poljak PhD , Mario Cvetković PhD , in Human Interaction with Electromagnetic Fields, 2019

2.1.3 The Continuity Equation

The equation of continuity couples the electromagnetic field sources (the charges and current densities) and can be readily derived from Maxwell equation (2.4). Taking the divergence of the Maxwell equation (2.4) yields

(2.23) $\mathrm{\nabla }\cdot (\mathrm{\nabla }\times \overrightarrow{H})=\mathrm{\nabla }\cdot \overrightarrow{J}+\mathrm{\nabla }\cdot (\frac{\partial \overrightarrow{D}}{\partial t}).$

As the left-hand side of (2.23) vanishes identically, it follows that

(2.24) $\mathrm{\nabla }\cdot \overrightarrow{J}+\frac{\partial }{\partial t}(\mathrm{\nabla }\cdot \overrightarrow{D})=0.$

Invoking Gauss law (2.5), the equation of continuity is obtained, namely

(2.25) $\mathrm{\nabla }\cdot \overrightarrow{J}=-\frac{\partial \rho }{\partial t},$

which for time-harmonic dependencies simplifies into

(2.26) $\mathrm{\nabla }\cdot \overrightarrow{J}=-j\omega \rho .$

The rate of charge moving out of a region is equal to the time rate of charge density decrease. The integral form of the continuity equation is obtained by performing the volume integration

(2.27) $\underset{V}{\int }\mathrm{\nabla }\cdot \overrightarrow{J}dV=-\frac{\partial }{\partial t}\underset{V}{\int }\rho dV$

and applying the Gauss divergence theorem

(2.28) $\underset{V}{\int }\mathrm{\nabla }\cdot \overrightarrow{J}dV=\underset{S}{\oint }\overrightarrow{J}\cdot d\overrightarrow{S}.$

The integral form of the continuity equation is then given by

(2.29) $\underset{S}{\oint }\overrightarrow{J}\cdot d\overrightarrow{S}=-\frac{\partial Q}{\partial t},$

where the unit normal in $d\overrightarrow{S}$ is the outward-directed normal, and Q is the total charge within the volume

(2.30) $Q=\underset{V}{\int }\rho dV.$

Eq. (2.29) represents the Kirchhoff conservation law widely used in the circuit theory.

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Fluid Dynamics

NICOLAE CRACIUNOIU , BOGDAN O. CIOCIRLAN , in Mechanical Engineer's Handbook, 2001

1.13.6 EQUATION OF CONTINUITY

The equation of continuity is obtained from the principle of conservation of mass. For steady flow, the principle of conservation of mass becomes

(1.32) ${\rho }_1{A}_1{V}_1={\rho }_2{A}_2{V}_2=\text{const},$

or

(1.33) ${\gamma }_1{A}_1{V}_1={\gamma }_2{A}_2{V}_2,$

that is, the mass of fluid passing all sections in a stream of fluid per unit time is the same. If the fluid is incompressible (γ₁ = γ₂), Eq. (1.33) yields

(1.34) $Q=A_1{V}_1=A_2{V}_2=\text{const},$

where A ₁ and A ₂ are the cross-sectional areas of the stream at sections 1 and 2, respectively, and V ₁ and V ₂ are respectively the velocities of the stream at the same sections. Commonly used units of flow are cubic feet per second (cfs), gallons per minute (gpm), or million gallons per day (mgd).

For steady two-dimensional incompressible flow, the continuity equation is

(1.35) $A_{n_1}{V}_1=A_{n_2}{V}_2=A_{n_3}{V}_3=\text{const},$

where A_n terms are the areas normal to the respective velocity vectors.

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Modelling of Turbulent Flows within Plant/Urban Canopies

H. HIRAOKA , in Computational Wind Engineering 1, 1993

(equation of continuity)

$\frac{1}{G}\frac{\partial G〈{\overline{U}}_j〉}{\partial x_j}=0$

(equation of momentum)

$\frac{\partial 〈{\overline{U}}_i〉}{\partial t}+\frac{1}{G}\frac{\partial G〈{\overline{U}}_k{\overline{U}}_i〉}{\partial x_k}=-\frac{1}{G}\frac{\partial G〈\overline{P}〉}{\partial x_1}-\frac{1}{G}\frac{\partial G\overline{u_i{u}_k}}{\partial x_k}-F{r}_i$

(equation of Reynolds stress)

$\frac{\partial \overline{u_i{u}_j}}{\partial t}+\frac{1}{G}\frac{\partial G〈{\overline{U}}_k〉\overline{u_i{u}_j}}{\partial x_k}=P_{ij}+F_{ij}+{\Pi }_{ij}-\frac{2}{3}\epsilon {\delta }_{ij}+D_{ij}$

(equation of energy dissipation)

$\frac{\partial \epsilon }{\partial t}+\frac{1}{G}\frac{\partial G〈\overline{U_k}〉\text{ε}}{\partial x_k}=\left(\frac{\text{ε}}{k}\right)\lfloor C_{\text{1ε}}P+C_{\text{pε}}{F}_{\text{ε}}-C_{2\text{ε}}\text{ε}\rfloor +D_{\text{ε}}$

, where

$\begin{array}{l}{P}_{ij}=-\frac{\overline{u_i{u}_k}}{G}\frac{\partial G\left(\overline{U_j}\right)}{\partial x_k},P=\frac{1}{2}{P}_{ij}\\ {\Pi }_{ij}={\varphi }_{ij}^1+{\varphi }_{ij}^2+{\varphi }_{ij}^{w1}+{\varphi }_{ij}^{w2}\\ {\varphi }_{ij}^1=-C_{1\pi }\left(\frac{\text{ε}}{k}\right)\left[\overline{u_i{u}_j}-\frac{2}{3}k{\text{δ}}_{ij}\right],{\varphi }_{ij}^2=-C_{2\pi }\left[P_{ij}-\frac{2}{3}P{\text{δ}}_{ij}\right]\\ {\varphi }_{ij}^{w1}=C_{\text{1π}}^{\text{'}}\left(\frac{\text{ε}}{k}\right)\left[\overline{u_k{u}_m}{n}_k{n}_m{\text{δ}}_{ij}-\frac{3}{2}\overline{u_k{u}_i}{n}_k{n}_i-\frac{3}{2}\overline{u_k{u}_j}{n}_k{n}_j\right]\frac{k^{\frac{3}{2}}}{C_w{x}_n\text{ε}}\\ {\varphi }_{ij}^{w2}=C_{\text{2π}}^{\text{'}}\left[{\varphi }_{km}^2{n}_k{n}_m{\text{δ}}_{ij}-\frac{3}{2}{\varphi }_{ik}^2{n}_k{n}_j-\frac{3}{2}{\varphi }_{jk}^2{n}_k{n}_i\right]\frac{k^{\frac{3}{2}}}{C_w{x}_n\text{ε}}\\ F_{ij}=〈{\overline{U}}_j〉F{r}_i+〈\overline{U_i}〉F{r}_j,D_{\text{ε}}=\frac{C_{\text{ε}}}{G}\frac{\partial }{\partial x_k}\left[\left(\frac{k}{\text{ε}}\right)\overline{u_k{u}_l}\frac{\partial G\text{ε}}{\partial x_l}\right]\\ D_{ij}=\frac{1}{G}\frac{\partial }{\partial x_k}\left[CsCrr\left(\frac{k}{\text{ε}}\right)\overline{u_k{u}_l}\frac{\partial G\overline{u_i{u}_j}}{\partial x_l}\right],F_{\text{ε}}=\frac{k^{\frac{3}{2}}}{L}\end{array}$

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Annular- and Leakage-Flow-Induced Instabilities

Michael P. Païdoussis , in Fluid-Structure Interactions (Second Edition), 2016

7.5.1 Hobson et al. studies

Hobson's (1982) seminal study of annular-flow-induced instabilities had a defining influence on all subsequent work on this subject, including that on plate-like structures in 1-D flow (Section 7.4). Hobson's work was motivated by the annular-flow-related problems in AGRs mentioned in Section 7.1.2.

The system under consideration is shown in Figure 7.26 (a). It consists of a rigid centrebody in a narrow annulus, both uniform except near the ends. Various end conditions are considered, as shown in Figure 7.26 (b). What follows, is based on an amalgam of Hobson's (1982) and Spurr & Hobson's (1984) work. The main assumptions are that (i) the annulus is very thin (H/a << 1, where r = a is the mean radius), so that variations of fluid variables in the radial direction can be neglected, (ii) the perturbation velocities are small compared to the mean axial flow velocity, allowing linearization, and (iii) the dynamic relationships between pressure, velocity and annular clearance may be obtained by linearization of the steady-state relationships.

Figure 7.26. A cylindrical centrebody in annular flow: (a) system definition; (b) different configurations, "a", "b", "c" and "d", of inlet and outlet (Hobson, 1982).

The equations of continuity and momentum may be derived by considering the mass and momentum fluxes into and out of a small annular element, and the frictional forces on the walls, yielding

(7.61) $\begin{array}{ll}\hfill \frac{\partial h}{\partial t}& +\frac{\partial (hu)}{\partial x}+\frac{1}{r}\frac{\partial (hv)}{\partial \theta }=0,\hfill \end{array}$

(7.62) $\begin{array}{ll}\hfill \frac{\partial }{\partial t}& (\rho hu)+\frac{\partial }{\partial x}\left[h(p+\rho u^2)\right]+\frac{1}{r}\frac{\partial }{\partial \theta }(\rho huv)-p\frac{\partial h}{\partial x}+2{\tau }_x=0,\hfill \end{array}$

(7.63) $\begin{array}{ll}\hfill \frac{\partial }{\partial t}& (\rho hv)+\frac{\partial }{\partial x}\left(\rho huv\right)+\frac{1}{r}\frac{\partial }{\partial \theta }\left[h\left(p+\rho v^2\right)\right]-\frac{p}{r}\frac{\partial h}{\partial \theta }+2{\tau }_{\theta }=0,\hfill \end{array}$

† where u and v are the axial and circumferential flow velocities, ρ the fluid density, p the pressure, and τ _x and τ _θ the frictional forces, defined by

(7.64) $\begin{array}{l}\hfill {\tau }_x=\tau (u/V),{\tau }_{\theta }=\tau (v/V),\tau =C_f\frac{1}{2}\rho V^2,V^2=u^2+v^2;\end{array}$

an average value for C _f is obtainable from pressure-gradient measurements in steady flow, using H(dP/dx) = −C _f ρ U ², in which U is the steady flow velocity.

Assuming planar centrebody motions in the θ = 0 plane, we can write

(7.65) $\begin{array}{l}\hfill h(x,\theta ,t)=H+h^{\prime }(x)cos\theta {\text{e}}^{\text{i}\mathrm{\Omega }t},\end{array}$

where the prime denotes the perturbed quantity. Hence, we express

(7.66) $\begin{array}{l}\hfill \begin{array}{cc}u(x,\theta ,t)=U+u^{\prime }(x)cos\theta {\text{e}}^{\text{i}\mathrm{\Omega }t},& v(x,\theta ,t)=v^{\prime }(x)sin\theta {\text{e}}^{\text{i}\mathrm{\Omega }t},\\ p(x,\theta ,t)=P(x)+p^{\prime }(x)cos\theta {\text{e}}^{\text{i}\mathrm{\Omega }t},& {\tau }_x={\overline{\tau }}_x+{\tau }_x^{\prime },{\tau }_{\theta }={\overline{\tau }}_{\theta }+{\tau }_{\theta }^{\prime }.\end{array}\end{array}$

Furthermore, unsteady skin friction terms are postulated to be obtainable by linearization of equation [7.64], e.g. ${\tau }_x^{\prime }=C_f{\left[\frac{1}{2}\rho V^2(u/V)\right]}^{\prime }\simeq C_f\rho U{u}^{\prime }$ ; thus,

(7.67) $\begin{array}{l}\hfill {\tau }_x^{\prime }=C_f\rho U{u}^{\prime },{\tau }_{\theta }^{\prime }=\frac{1}{2}{C}_f\rho U{v}^{\prime }.\end{array}$

Substitution of equations [7.65]–(7.67) into (7.61)–(7.63), linearization, and elimination of the steady flow equations, leads to a set of linear first-order equations in $u^{\prime }(x),v^{\prime }(x)$ and $p^{\prime }(x)$ :

(7.68) $\begin{array}{ll}\hfill & \frac{\text{d}{u}^{\prime }}{\text{d}x}+\frac{1}{a}{v}^{\prime }=-\left(\frac{U}{H}\frac{\text{d}}{\text{d}x}+\frac{\text{i}\mathrm{\Omega }}{H}\right)h^{\prime },\hfill \\ \hfill & \rho \left(\text{i}\mathrm{\Omega }+2U\frac{\text{d}}{\text{d}x}+2C_f\frac{U}{H}\right)u^{\prime }+\frac{\rho U}{a}{v}^{\prime }+\frac{\text{d}{p}^{\prime }}{\text{d}x}=\hfill \\ \hfill & -\left(\text{i}\mathrm{\Omega }\rho \frac{U}{H}+\frac{\rho U^2}{H}\frac{\text{d}}{\text{d}x}+\frac{1}{H}\frac{\text{d}P}{\text{d}x}\right)h^{\prime },\hfill \\ \hfill & \rho \left(\text{i}\mathrm{\Omega }+U\frac{\text{d}}{\text{d}x}+C_f\frac{U}{H}\right)v^{\prime }-\frac{1}{a}{p}^{\prime }=0,\hfill \end{array}$

in which a is the mean radius of the annulus.

Various fluid boundary conditions are considered at the inlet and outlet, as follows. (i) Short loss-less inlet. The flow accelerates rapidly over a very short distance from a reservoir, where the pressure is P ₀ and constant, up to the annulus inlet at x = 0, so that $p+\frac{1}{2}\rho (u^2+v^2)=P_0$ . If the incoming flow is irrotational, then $v^{\prime }(0)=0$ . Hence, at the inlet

(7.69) $\begin{array}{l}\hfill p^{\prime }(0)+\rho U{u}^{\prime }(0)=0.\end{array}$

(ii) A constricted inlet or outlet. At the throat of the constriction the mean width is

(7.70) $\begin{array}{l}\hfill H^{*}=\mu H.\end{array}$

An asterisk denotes 'at the constriction' throughout. It is assumed that h* varies with vibration of the centrebody, $h(t)=H+h^{\prime }(t)$ , such that $h^{*}(t)=H^{*}+h^{\prime }(t)$ . Also, by continuity, we have u* h* = uh. By Bernoulli's equation, we may write $p^{*}+\frac{1}{2}\rho (u^{*2}+v^{*2})=$ const. Combining the foregoing and linearizing, we have at x = 0

(7.71) $\begin{array}{l}\hfill p^{\prime }(0)=-\frac{\rho U^2}{\mu }\left(\frac{h^{\prime }}{H^{*}}-\frac{h^{\prime }}{\mu H^{*}}+\frac{u^{\prime }}{\mu U}\right){|}_0.\end{array}$

In a similar way for a constricted outlet, we have

(7.72) $\begin{array}{l}\hfill u^{\prime }(L)={\left[\frac{U^{*}}{H^{*}(1+\mu )}{h}^{\prime }+\frac{\mu }{(1-{\mu }^2)U^{*}}{p}^{\prime }\right]}_L.\end{array}$

(iii) Free discharge. In this case the perturbation at x = L is simply

(7.73) $\begin{array}{l}\hfill p^{\prime }(L)=0.\end{array}$

(iv) Outlet variable-efficiency diffuser. Although it is recognized that the performance of the diffuser is generally dependent on θ as the centrebody is displaced, a quasi-steady, quasi-one-dimensional approach is taken, in which the pressure recovery from the annulus pressure p to the constant static pressure in the downstream reservoir $p_{\infty }$ is expressed as

(7.74) $\begin{array}{l}\hfill p_{\infty }-p=\frac{1}{2}\rho u^2\eta (h),\end{array}$

where η(h) is the pressure recovery factor. Further, assuming

$\eta (h)=\overline{\eta }+{\eta }^{\prime },{\eta }^{\prime }=\left[\text{d}\eta (h)/\text{d}h\right]h^{\prime },$

equation [7.74] is linearized to give

(7.75) $\begin{array}{l}\hfill p^{\prime }(L)=-\frac{1}{2}\rho U^2{\eta }^{\prime }-\eta \rho U{u}^{\prime }(L).\end{array}$

Furthermore, a diffuser performance parameter δ, of order unity, may be defined,

(7.76) $\begin{array}{l}\hfill \delta =\left[\text{d}\eta (h)/\text{d}h\right]H.\end{array}$

For any given configuration, $\overline{\eta }$ may be determined from steady-state experiments with the centrebody concentric; δ, however, must be measured from several ad hoc experiments in which the centrebody is moved, so that the flow passage becomes progressively more eccentric.

The linearized equations [7.68], together with the appropriate boundary conditions upstream and downstream are sufficient to define the flow problem. They may be solved by first eliminating $v^{\prime }$ and then solving the resulting equations in $u^{\prime }$ and $p^{\prime }$ by a finite difference scheme using central differences for all derivatives (Spurr & Hobson, 1984). At the boundaries, second-order accuracy may be retained by eliminating the axial velocity algebraically and using instead a second-order Poisson equation for $p^{\prime }$ . This results in a block tridiagonal matrix relating $u^{\prime }$ and $p^{\prime }$ at discrete axial locations to the values of $h^{\prime }$ , which may be solved by standard means.

Hobson (1982), however, presents analytical solutions ² by neglecting τ _x and τ _θ in equations [7.62] and (7.63) and simplifying $h(x)=\overline{h}$ . The solution consists of complementary and particular parts,

$u^{\prime }(x)=\sum _{i=1}^3{\overline{u}}_i{\text{e}}^{{\alpha }_{i}x},v^{\prime }(x)=\sum _{i=1}^3{\overline{v}}_i{\text{e}}^{{\alpha }_{i}x}-\frac{\text{i}\mathrm{\Omega }a}{H}\overline{h},p^{\prime }(x)=\sum _{i=1}^3{\overline{p}}_i{\text{e}}^{{\alpha }_{i}x}+\frac{\rho {\mathrm{\Omega }}^2{a}^2}{H}\overline{h}.$

Substituting into equation [7.68] one obtains a characteristic equation for α _i and hence three solutions for ${\overline{u}}_i,{\overline{v}}_i$ and ${\overline{p}}_i$ . The boundary conditions are similarly expressed in terms of ${\overline{u}}_i$ and ${\overline{p}}_i$ .

In either form of the solution, the problem is completed by the modal equation for the structure (the centrebody), associated with $h(x,\theta ,t)=h^{\prime }(t)\psi (x,\theta )$ :

(7.77) $\begin{array}{l}\hfill \frac{{\text{d}}^2{h}^{\prime }}{\text{d}{t}^2}+2{\zeta }_s{\mathrm{\Omega }}_s\frac{\text{d}{h}^{\prime }}{\text{d}t}+{\mathrm{\Omega }}_s^2{h}^{\prime }=F^{\prime }/M_s,\end{array}$

where subscript s stands for 'structure', and

$F^{\prime }(t)=a{\int }_0^L{\int }_0^{2\pi }{p}^{\prime }(x,\theta ,t)\psi (x,\theta )\text{d}\theta \text{d}t,M_s=a{\int }_0^L{\int }_0^{2\pi }m(x,\theta ){\psi }^2(x,\theta )\text{d}\theta \text{d}x;$

m(x,θ) being the mass of the structure per unit annular area. For $h^{\prime }(t)=\overline{h}exp(\text{i}\mathrm{\Omega }t)$ , $F^{\prime }(t)=\overline{F}exp(\text{i}\mathrm{\Omega }t)$ , the added mass M _a and added damping ζ _a are given by

(7.78) $\begin{array}{l}\hfill M_a=\mathcal{R}e(\overline{F})/({\mathrm{\Omega }}^2\overline{h}),2\mathrm{\Omega }{\zeta }_a=-\mathcal{I}m(\overline{F})/(\mathrm{\Omega }{M}_s\overline{h}).\end{array}$

Sample results of the dimensionless fluid-dynamic damping factor, ${\zeta }_a^{*}$ , as a function of the reduced flow velocity, $\overline{U}$ , defined by

(7.79) $\begin{array}{l}\hfill {\zeta }_a=\left(\rho a^3/M_s\right)(a/H){\zeta }_a^{*},\overline{U}=U/(\mathrm{\Omega }a),\end{array}$

are shown in Figures 7.27 and 7.28. For all the cases to be discussed, Hobson gives analytical expressions for ${\zeta }_a^{*}$ ; e.g. for an upstream constriction,

Figure 7.27. The fluid-dynamic damping factor ${\zeta }_a^{*}$ versus reduced flow velocity $\overline{U}$ , defined in equation [7.79]. (a,b) The effect of an upstream constriction, characterized by μ: (a) for ε = 1; (b) for $\epsilon \to \infty$ . (c) The effect of a downstream constriction (Hobson, 1982).

Figure 7.28. The effect on the fluid-dynamic damping of a downstream diffuser with a gradually improving diffuser efficiency, δ = 0.1 (Hobson, 1982).

(7.80) $\begin{array}{l}\hfill {\zeta }_a^{*}=\frac{\pi }{2\omega }\left(1-\text{sech}\epsilon \right)\left\{\frac{2\mu -1+\mu {\omega }^2(1-\text{sech}\epsilon )}{1+{(\mu \omega tanh\epsilon )}^2}\right\},\end{array}$

where ω = Ωa/U*, U* = U/μ being the 'throat' velocity and μ is defined in equation [7.70], and ε = L/a; for a downstream constriction,

(7.81) $\begin{array}{l}\hfill {\zeta }_a^{*}=\frac{\pi }{2\omega }(1-\text{sech}\epsilon )\left\{\frac{\mu {\omega }^2(1-\text{sech}\epsilon -{\mu }^2(1-\mu )(1+\text{sech}\epsilon )+1}{1+{(\mu \omega tanh\epsilon )}^2}\right\}.\end{array}$

Figure 7.27 (a) shows the effect of an upstream constriction for a short annulus, ε = L/a = 1; a smaller μindicates a larger constriction, while μ = 1 corresponds to the datum configuration 'a' of Figure 7.26 (b) (no constriction). It is seen that the datum configuration remains stable with increasing $\overline{U}$ . The same is true for other upstream constrictions with μ ≥ 0.5. For μ < 0.5, however, negative damping is generated, increasingly more pronounced for smaller μ; the critical $\overline{U}$ at which ${\zeta }_a^{*}<0,{\overline{U}}_{\text{cr}}$ , is correspondingly lower. For a long annulus, $\epsilon \to \infty$ , the results, in Figure 7.27 (b), are similar. However, the effect is not uniform; e.g. for μ = 0.4, for the higher $\epsilon ,{\overline{U}}_{\text{cr}}$ is higher, but ${\zeta }_a^{*}$ is more strongly negative thereafter.

For a downstream constriction, however, the system remains stable for all μ, as seen in Figure 7.27 (c). Thus, Miller's (1970) findings – see Section 7.1.2 – are fully reproduced, establishing more firmly still the desirability of placing constrictions downstream, for stability.

Annuli with a datum-configuration inlet and a diffuser outlet are considered next. If the diffuser efficiency is assumed not to vary with h, i.e. δ = 0, increasing efficiency η is found to reduce the positive damping of the datum configuration until, at 100% pressure recovery, i.e. η = 1, the aerodynamic damping vanishes. If the efficiency deteriorates with h, e.g. δ = −1, the aerodynamic damping remains positive even with η = 1. If, on the other hand, efficiency improves with h, e.g. δ = 1, the system may become unstable for a high enough pressure recovery (high η) at sufficiently high $\overline{U}$ , as shown in Figure 7.28. Hence, a good diffuser may by itself be responsible for instability.

Some of these theoretical findings were tested by experiments in the apparatus shown in Figure 7.29. The centrebody is made of a 25.4   mm steel tube with 0.25   mm wall thickness, purposely light for inertia not to overwhelm the fluid forces in the measurements. The diametral gap was only 1   mm. The diffuser was of 6^o half-angle, but could be replaced by a straight section to reproduce the datum configuration, or by diffusers of different half-angle.

Figure 7.29. The apparatus used by Hobson et al. for the dynamics of a cylindrical centrebody in annular flow (Spurr &amp; Hobson, 1984).

Theory and experiment are compared in Figure 7.30: (a) for the effect of an upstream constriction (no diffuser), and (b, c) for the effect of a diffuser (no upstream constriction). It is seen in Figure 7.30 (a) that theory and experiment agree very well for the datum configuration. With the upstream constriction, however, agreement is less good: although the instability and the qualitative observed behaviour are correctly predicted, theory overpredicts the threshold.

Figure 7.30. (a) Comparison between theory and experiment for the case of no constriction (μ = 1, datum) or an upstream constriction (μ = 0.18); (Hobson, 1982). (b,c) The real and imaginary parts of the normalized force/displacement transfer function, g(ω), versus reduced frequency, ω for (b) 4^o diffuser half-angle, and (c) 6^o half-angle (Spurr &amp; Hobson, 1984).

In the experiments on the effect of the diffuser (Spurr & Hobson, 1984), the force/displacement transfer function was measured directly, with an impedance head on the shaker, using random noise excitation over a frequency range of 20–120 Hz. In the results presented in Figure 7.30 (b,c), g(ω) = X _a H/(a ²Δp), where X _a is the aerodynamic transfer function, obtained by subtracting the flow-off from the flow-on value, and a ²Δp is a fraction of the force on the centrebody undergoing a vibration taking up the full clearance, Δp being the pressure drop across the annulus. A reference flow velocity is then defined by $\mathrm{\Delta }p=\frac{1}{2}\rho U_{\text{ref}}^2$ , which is used to evaluate the reduced frequency, ω = Ωa/U _ref. The real part of $g(\omega ),\mathcal{R}e(g)$ , represents the fluid stiffness, and the imaginary part, $\mathcal{I}m(g)$ , the fluid damping.

It is seen in Figure 7.30 (b,c) that for both diffuser half-angles of 4^o and 6 ${}^{\text{o}},\mathcal{R}e(g)<0$ . For the force and displacement sign conventions utilized by Spurr & Hobson, this signifies a positive fluid stiffness, hence a centralizing force on the centrebody. In Figure 7.30 (b), we also have $\mathcal{I}m(g)<0$ , which for the sign convention used means a positive damping; in Figure 7.30 (c), however, for the 6^o diffuser, $\mathcal{I}m(g)>0$ , signifying a negative aerodynamic damping, and hence the possibility of self-excited vibrations. This, indeed, is what was observed in the experiments with varying flow.

Other results, for 0^o and 2^o diffusers show similar agreement as in Figure 7.30 (b). It should be remarked, however, that in the theoretical calculations the value of η used was approximately double the value obtained by steady-state measurements. This straining of parameters (cf. Section 5.8.1¹) does not invalidate the comparison; nevertheless, quantitative agreement is not as good as shown.

This work has been extended by Hobson (1984) to include the effects of arbitrary structural mode shape and arbitrarily nonuniform annular geometry. Further work on the effect of diffusers has been done by Parkin & Watson (1984), Parkin et al. (1987) and Hobson & Jedwab (1990). In the latter, it is pointed out that, provided the centrebody and the channel are coaxial, empirical measurement of loss and recovery factors associated with sudden expansions permit the type of theory just described to deal with stability in such configurations. Even this, however, is not feasible for eccentrically located centrebodies in diffusers. The flow then becomes very complex. Intriguingly, for a 30^o diffuser, a Strouhal periodicity may exist in the flow, even in the absence of movement; with motion, its continued existence is dependent on vibration amplitude and frequency in a complicated manner (Hobson & Jedwab, 1990). However, this is beyond the scope of this discussion.

Hobson's and Spurr & Hobson's findings have been reconfirmed by analysis and a set of very nice water-flow experiments by Fujita & Ito (1992).

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Airfoils and Wings in Compressible Flow

E.L. Houghton , ... Daniel T. Valentine , in Aerodynamics for Engineering Students (Seventh Edition), 2017

The Equations of Motion of a Compressible Fluid

The equation of continuity may be recalled in Cartesian coordinates for two-dimensional flow in the form

(8.1) $\frac{\partial \rho }{\partial t}+\frac{\partial (\rho u)}{\partial x}+\frac{\partial (\rho v)}{\partial y}=0$

since, in what follows, analysis of two-dimensional conditions is sufficient to demonstrate the method and derive valuable equations. The equations of motion may also be recalled in similar notation as

(8.2) $\begin{array}{c}\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=-\frac{1}{\rho }\frac{\partial p}{\partial x}\hfill \\ \frac{\partial v}{\partial t}+u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}=-\frac{1}{\rho }\frac{\partial p}{\partial y}\hfill \end{array}\}$

and, for steady flow,

(8.3) $\begin{array}{c}-\frac{1}{\rho }\frac{\partial p}{\partial x}=u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}\hfill \\ -\frac{1}{\rho }\frac{\partial p}{\partial y}=u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}\hfill \end{array}\}$

For adiabatic flow (since the assumption of negligible viscosity has already been made, further stipulations of adiabatic compression and expansion imply isentropic flow),

(8.4) $p=k{\rho }^{\gamma },\frac{\partial p}{\partial \rho }=a^2=\frac{\gamma p}{\rho }$

For steady flow, Eq.(8.1) may be expanded to

(8.5) $u\frac{\partial \rho }{\partial x}+v\frac{\partial \rho }{\partial y}+\rho \frac{\partial u}{\partial x}+\rho \frac{\partial v}{\partial y}=0$

but

$\frac{\partial \rho }{\partial x}=\frac{\partial p}{\partial x}\frac{\partial \rho }{\partial p}=\frac{1}{a^2}\frac{\partial p}{\partial x},\text{etc.}$

so Eq.(8.5) becomes

(8.6) $\frac{u}{a^2}\frac{\partial p}{\partial x}+\frac{v}{a^2}\frac{\partial p}{\partial y}+\rho \frac{\partial u}{\partial x}+\rho \frac{\partial v}{\partial y}=0$

Substituting in Eq.(8.6) for $\partial p/\partial x$ , $\partial p/\partial y$ from Eq.(8.3) and canceling ρ gives

$-\frac{u^2}{a^2}\frac{\partial u}{\partial x}-\frac{uv}{a^2}\frac{\partial u}{\partial y}-\frac{vu}{a^2}\frac{\partial v}{\partial x}-\frac{v^2}{a^2}\frac{\partial v}{\partial y}+\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0$

or, collecting like terms,

(8.7) $(1-\frac{u^2}{a^2})\frac{\partial u}{\partial x}-\frac{uv}{a^2}(\frac{\partial v}{\partial x}+\frac{\partial u}{\partial y})+(1-\frac{v^2}{a^2})\frac{\partial v}{\partial y}=0$

For irrotational flow $\partial v/\partial x=\partial u/\partial x$ , and a velocity potential ${\phi }_1$ (say) exists, so

(8.8a) $(1-\frac{u^2}{a^2})\frac{\partial u}{\partial x}-\frac{2uv}{a^2}\frac{\partial u}{\partial y}+(1-\frac{v^2}{a^2})\frac{\partial v}{\partial y}=0$

and since $u=\partial {\phi }_1/\partial x,v=\partial {\phi }_1/\partial y$ , Eq.(8.8a) can be written as

(8.8b) $(1-\frac{u^2}{a^2})\frac{{\partial }^2{\varphi }_1}{\partial x^2}-\frac{2uv}{a^2}\frac{{\partial }^2{\varphi }_1}{\partial x\partial y}+(1-\frac{v^2}{a^2})\frac{{\partial }^2{\varphi }_1}{\partial y^2}=0$

Finally, the energy equation provides the relation between a, u, v, and acoustic speed. Thus

(8.9) $\frac{u^2+v^2}{2}+\frac{a^2}{\gamma -1}=\text{constant}$

or

(8.10) ${\left(\frac{\partial {\varphi }_1}{\partial x}\right)}^2+{\left(\frac{\partial {\varphi }_1}{\partial y}\right)}^2+\frac{2}{\gamma -1}{a}^2=\text{constant}$

Combining Eqs.(8.8b) and(8.10) gives an expression in terms of the local velocity potential.

Even without continuing the algebra beyond this point, we note that the resulting nonlinear differential equation in ${\phi }_1$ is not amenable to a simple closed solution and that further restrictions on the variables are required. Since all possible restrictions on the generality of the flow properties have been made, it is necessary to consider the component velocities themselves.

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Large-Scale UAV Trajectory Planning Using Fluid Dynamics Equations

Mohammadreza Radmanesh , ... Manish Kumar , in Multi-Rotor Platform-based UAV Systems, 2020

4.4 Fluid analogy

The continuity equation of fluid mechanics is used as the basis function f for generating all of the possible path solutions in the porous domain $\mathrm{ℂ}$ . This is represented in equation [4.3], where ϕ is the porosity of the evaluated three-dimensional point, ρ is the density of the fluid and u is the velocity of the fluid:

[4.3] ${\left(\mathit{\varphi \rho }\right)}_t+\nabla .\left(\mathit{\rho u}\right)=f\text{in}\mathrm{ℂ},$

Then, by applying Darcy's law to equation [4.4], we describe the flow of the fluid through the porous domain. The steady-state solution of equation [4.3] is applied to obtain:

[4.4] $-\sum _{e\in \left\{x,y,z\right\}}\left[\frac{\partial }{\partial e}\kappa \frac{\partial u}{\partial e}\right]=f$

In equation [4.4], κ represents the permeability of the medium $\mathrm{ℂ}$ . It should be noted that due to the finite element method for particles in fluid, the computational domain always deals with real numbers. The second-order centered difference method is applied to each spatial point that interpolates the fluid particle properties using only the values of the cell before and after the currently evaluated cell in order to approximate the porosity of each cell. To do this, equation [4.4] is discretized in the x, y and z directions using the finite difference method (Mitchell and Griffiths 1980). The substitution into [4.4] results in [4.5]. The indices in equation [4.5] represent the grid points. The variables h _x , h _y and h _z are the step sizes in the x, y and z directions respectively. They are specifically defined differently because the x, y and z grid intervals are not equidistant. O(h _e ²) is the error of numerical estimation that can be neglected (Ferziger and Peric 2012) by considering a small grid size h _e :

[4.5] $\begin{array}{l}-\frac{{\kappa }_{i+1/2,j,k}\left(u_{i+1,j,k}-u_{i,j,k}\right)-{\kappa }_{i-1/2,j,k}\left(u_{i,j,k}-u_{i-1,j,k}\right)}{h_x^2}\\ -\frac{{\kappa }_{i,j+1/2,k}\left(u_{i,j+1,k}-u_{i,j,k}\right)-{\kappa }_{i,j-1/2,k}\left(u_{i,j,k}-u_{i,j-1,k}\right)}{h_y^2}\\ -\frac{{\kappa }_{i,j,k+1/2}\left(u_{i,j,k+1}-u_{i,j,k}\right)-{\kappa }_{i,j,k-1/2}\left(u_{i,j,k}-u_{i,j,k-1}\right)}{h_z^2}\\ =f_{i,j,k}\end{array}$

The variables (i,j,k) in equation [4.5] represent the cell labels of the tessellated area. Equation [4.5] is solved for the entire domain $\mathrm{ℂ}$ while accounting for the boundary conditions.

The results of equation [4.5] show multiple streamlines with varying fluid velocity u at each grid point. Since there is only one starting point and one final point in the domain, the number of solutions is reduced by applying the flux boundary conditional constraints which require that solutions start at time "t ₀" at (x ₀, y ₀, z ₀) and finish at time " $\mathbb{T}$ " at (x _f , y _f , z _f ). The process can be explained as follows:

In the domain $\mathrm{ℂ}$ , multiple streamlines or flow paths are generated from the original position to the goal position. Incompressible flow dynamics represented by equation [4.5] dictate that the streamlines follow the velocity fields and finish at the goal position. The streamlines are evaluated with the cost equation (i.e. safely reaching to the goal position with the shortest distance traveled) and are converted to arrays of (x, y, z) points using the Runge–Kutta method (RKM) algorithm (Radmanesh et al. 2017). The RKM yields all the routes that can be used to travel from the initial point to the goal position without colliding with any other agents (i.e. the solid part of the computational domain $\mathrm{ℂ}$ ) and violating the fluid constraints. Further studies on this method are provided clearly in (Radmanesh et al. 2017).

The streamlines, generated from the technique explained above, contain spatio-temporal information about the fluid particle. In the following section, we will use this characteristic of the solution to generate the prediction sets for UAVs. Of course, only the streamlines that satisfy the UAV kinematic and dynamic constraints will be used to generate the paths.

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