Continuity for Incompressible Inviscid Flows Implies That as the Cross Sectional Area is Reduced

Incompressible Flow

Accurate incompressible N-S solution on cluster of work stationsa.

A.R. Aslan , ... A. Misirlioglu , in Parallel Computational Fluid Dynamics 1998, 1999

Viscous incompressible flows past complex shapes are studied with accurate solution to Navier-Stokes equations in cluster of workstations operating in PVM environment. The Finite Element Method with a two-step explicit time marching scheme is used for solution of the momentum equations. The Domain Decomposition together with the element-by-element (EBE) iteration technique is employed for solution of the auxiliary potential function. Matching and nonoverlapping grids are used for solution of both the momentum and the auxiliary potential equations. The cubic cavity problem is solved for 2, 4 and 6 domains having equal number of grid points. The solid super-linear speed-up obtained on the cluster is very satisfying.

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Parallel Property of Pressure Equation Solver with Variable Order Multigrid Method for Incompressible Turbulent Flow Simulations

Hidetoshi Nishida , Toshiyuki Miyano , in Parallel Computational Fluid Dynamics 2006, 2007

1 INTRODUCTION

The incompressible flow simulations are usually based on the incompressible Navier-Stokes equations. In the incompressible Navier-Stokes equations, we have to solve not only momentum equations but also elliptic partial differential equation (PDE) for the pressure, stream function and so on. The elliptic PDE solvers consume the large part of total computational time, because we have to obtain the converged solution of this elliptic PDE at every time step. Then, for the incompressible flow simulations, especially the large-scale simulations, the efficient elliptic PDE solver is very important key technique.

In the parallel computations, the parallel performance of elliptic PDE solver is not usually high in comparison with the momentum equation solver. When the parallel efficiency of elliptic PDE solver is 90% on 2 processor elements (PE)s. that is. the speedup based on 1PE is 1.8. the speedup on 128PEs is about 61 times of 1PE. This shows that we use only the half of platform capability. On the other hand, the momentum equation solver shows almost theoretical speedup [1], [2]. Therefore, it is very urgent problem to improve the parallel efficiency of elliptic PDE solver.

In this paper, the parallel property of elliptic PDE solver, i.e., the pressure equation solver, with variable order multigrid method [3] is presented. Also, the improvement of parallel efficiency is proposed. The present elliptic PDE solver is applied to the direct numerical simulation (DXS) of 3D turbulent channel flows. The message passing interface (MPI) library is applied to make the computational codes. These MPI codes are implemented on PRIME POWER, system with SPARC 64V (1.3GHz) processors at Japan Atomic Energy Agency (JAEA).

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Fully Coupled Solver for Incompressible Navier-Stokes Equations using a Domain Decomposition Method

Jerome Breil , ... Tadayasu Takahashi , in Parallel Computational Fluid Dynamics 2002, 2003

2.2 Initial and Boundary Conditions

We assume the initial condition

(5) $\mathbf{u}|{}_{\mathit{t}=0}={\mathbf{u}}_0\text{,}$

where ∇  ⋅ u ₀  =   0.

For incompressible flows the following boundary condition are usually imposed:

•

Prescribed velocity on inflow boundary;

•

No-slip conditions, zero normal stress on solid surface;

•

Prescribed gradient (usually zero) of all quantities on outflow surface;

•

Zero gradient normal to the boundary for all scalar quantities and the velocity components parallel to the surface on symmetry plane; zero velocity normal to such surface;

•

Transmission condition (to ensure the continuity of the velocity field and the continuity of the pressure or the normal stress across the interface) between two domains will exchange the data of each domain.

We can summarize the boundary conditions for the velocity, in the following way

•

Dirichlet boundary condition

(6) $\mathit{\phi }|{}_{\partial \mathit{\Omega }}={\mathit{\phi }}_{\mathrm{b}}\text{.}$

•

Neumann boundary condition

(7) ${\left.\frac{\partial \mathit{\phi }}{\partial \mathit{n}}\right|}_{\partial \mathit{\Omega }}=\mathit{q}\text{.}$

•

Mixed boundary condition

(8) ${\left.\frac{\partial \mathit{\phi }}{\partial \mathit{n}}\right|}_{\partial \mathit{\Omega }}=\mathit{\alpha }\left(\mathit{\phi }|{}_{\partial \mathit{\Omega }}-{\mathit{\phi }}_{\mathit{b}}\right)\text{.}$

Remark: In order to maintain compatibility, initial condition should satisfy boundary conditions at t = 0 and x ∈ ∂Ω

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Aircraft Speed and Altitude

William Gracey , ... Tony Whitmore , in Encyclopedia of Physical Science and Technology (Third Edition), 2003

II.B Airspeed Equations

In incompressible flow, the pressure developed by the forward motion of a body is called the dynamic pressure q, which is related to the true airspeed V by:

(10) $\text{q}=\frac{1}{2}\rho {\text{V}}^2$

where ρ is the density of the air and V the speed of the body relative to the air. Air, however, is compressible, and when airspeed is measured with a pitot–static tube, the air is compressed as it is brought to a stop in the pitot tube. As a consequence of this compression, the measured impact pressure is higher than the dynamic pressure. The effects of compressibility are taken into account in the following equations, in which the calibrated airspeed V _c is made equal to the true airspeed V at standard sea-level conditions:

(11) $\begin{array}{l}{\text{q}}_{\text{c}}={\text{p}}_0\left[{\left(1+\frac{\gamma -1}{2\gamma }\frac{{\rho }_0}{{\text{p}}_0}{\text{V}}_{\text{c}}^2\right)}^{\gamma /\left(\gamma -1\right)}-1\right]\hfill \\ {\text{V}}_{\text{c}}\underset{¯}{\underset{¯}{<}}{\text{a}}_0\hfill \end{array}$

(12) $\begin{array}{l}{\text{q}}_{\text{c}}=\frac{1-\gamma }{2}{\left(\frac{{\text{V}}_{\text{c}}}{{\text{a}}_{\text{0}}}\right)}^2{\text{p}}_0\times {\left[\frac{{\left(\gamma +1\right)}^2}{4\gamma -2\left(\gamma -1\right){\left({\text{a}}_0/{\text{V}}_{\text{c}}\right)}^2}\right]}^{1/\left(\gamma -1\right)}-{\text{p}}_0\hfill \\ {\text{V}}_{\text{c}}\underset{¯}{\underset{¯}{>}}{\text{a}}_0\hfill \end{array}$

where a is the speed of sound, γ is the ratio of the specific heats of air (with a value of 1.4), and the subscript 0 refers to values at sea level. Airspeed indicators are calibrated in accordance with Eq. (11) for subsonic speeds and Eq. (12) for supersonic speeds. Tables of impact pressure versus calibrated airspeed in both U.S. Customary Units and the International System of Units are given in Gracey (1981).

Since the airspeed indicator measures true airspeed only at standard sea-level conditions, the calibrated airspeeds at altitude will be lower than the true airspeeds, as shown by the following equation,

(13) $\text{V}-{\text{V}}_{\text{c}}\frac{\text{f}}{{\text{f}}_0}\sqrt{\frac{{\rho }_0}{\rho }}$

where ρ is the density and f a compressibility factor defined by:

(14) $\text{f}=\sqrt{\frac{\gamma }{\gamma -1}\frac{\text{p}}{{\text{q}}_{\text{c}}}\left[{\left(\frac{{\text{q}}_{\text{c}}}{\text{p}}+1\right)}^{\left(\gamma -1\right)/\gamma }-1\right]}$

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31st European Symposium on Computer Aided Process Engineering

M. Robles-Santacruz , ... A.R. Uribe-Ramírez , in Computer Aided Chemical Engineering, 2021

2 The SPH Numerical Model

For viscous incompressible flows, the governing equations are given by the Navier– Stokes equation expressed in SPH form as

(1) $\frac{d{\mathbf{v}}_a}{\mathit{dt}}=-\sum _b{m}_b\left(\frac{p_a+p_b}{{\rho }_a{\rho }_b}+\mathrm{\Gamma }\right){\mathbf{\nabla }}_a{W}_{\mathit{ab}}+\mathbf{g}$

where ρ is the density, p the pressure, v the velocity field, W is the kernel function, g is the gravity acceleration and Γ is the viscosity dissipative term. In a standard SPH formulation, where the pressure is given as a function of the density, local variations of the pressure gradient may induce local density fluctuations in the flow. Therefore, the flow is modelled by an artificial fluid that is approximately incompressible. The mass of a fluid element remains constant and only its associated density fluctuates. Such density fluctuations are calculated by solving the continuity equation expressed in the SPH in Eq. (2). The dynamical pressure p _d , which for simplicity we shall denote by p, is calculated using the relation expressed in Eq. (3) (Becker and Teschner, 2007).

(2) $\frac{d{\rho }_a}{\mathit{dt}}={\mathrm{\Sigma }}_{b=1}^N{m}_b\left({\mathbf{v}}_a-{\mathbf{v}}_b\right)\cdot {\mathbf{\nabla }}_a{W}_{\mathit{ab}},$

(3) $p=p_0\left[{\left(\frac{\rho }{{\rho }_0}\right)}^{\gamma }-1\right]$

where γ=7, p ₀ = c ₀ ² ρ ₀ /γ, ρ ₀ is a reference density, and c ₀ is the sound speed at the reference density. This equation enforces very low density fluctuations since the speed of sound can be artificially slowed with accurate results in fluid propagation. The viscous term in equation (1) is modelled using the artificial viscosity model proposed by Monaghan (1992) due to the simplicity and numerical stability. Finally, the surface tension is modelled according to Tartakovsky and Panchenko (2016) where a Pairwise-Force is inserted in the Eq. (1)

(4) $\frac{d{\mathbf{v}}_a}{\mathit{dt}}=-\sum _b{m}_b\left(\frac{p_a+p_b}{{\rho }_a{\rho }_b}+\mathrm{\Gamma }\right)\mathbf{\nabla }{\mathit{aW}}_{\mathit{ab}}+\mathit{g}+\frac{{\mathit{F}}_a}{m_a}$

where F _a is calculated from the interaction particles using the Eq. (6).

In Eqs. (4) the pressure gradient is written in SPH form using the symmetric representation proposed by Colagrossi and Landrini (2003), which ensures numerical stability at the interface between two media with large density differences, while the surface tension term F _a is calculated according to the sum of the F _ab formulations given by Tartakovsky and Panchenko (2016) in Eq. (6).

(5) $\mathrm{\Gamma }=\left\{\begin{array}{ll}\frac{-\alpha {\overline{c}}_{\mathit{ab}}{\mu }_{\mathit{ab}}}{{\rho }_{\mathit{ab}}}& {\mathbf{v}}_{\mathit{ab}}\cdot {\mathbf{r}}_{\mathit{ab}}<0\\ 0& {\mathbf{v}}_{\mathit{ab}}\cdot {\mathbf{r}}_{\mathit{ab}}\ge 0\end{array},\right.$

where ${\mu }_{\mathit{ab}}=h{\mathbf{v}}_{\mathit{ab}}\cdot {\mathit{r}}_{\mathit{ab}}/\left({{\mathit{r}}^2}_{\mathit{ab}}+{\eta }^2\right)$ , ${\overline{c}}_{\mathit{ab}}=0.5\left(c_a+c_b\right)$ , where c _a y c _b are the speed of sound for particle a and b respectively, η ²=0.01h ² and α is a free parameter that can be adjusted according to the simulation case.

(6) $F_{\mathit{ab}}=\left\{\begin{array}{ll}{S}_{\mathit{ab}}\mathit{cos}\left(\frac{1.5\pi }{3h}|r_b-r_a|\right)\frac{r_b-r_a}{\mid r_b-r_a\mid },& \mid r_b-r_a\mid \le h\\ 0,& \mid r_b-r_a\mid >h\end{array},\right.$

where the S _ab parameter can be calculated from the surface tension fluid, the smoothing length the kernel function and the contact angle between the wetting fluid and the bound as is reported by Tartakovsky and Panchenko (2016).

No-slip boundary conditions are implemented at the walls of the vessel using the method of dynamic boundary particles developed by Crespo et al. (2007). In this method, a linear distribution of uniformly-spaced particles is placed at the walls of the enclosure, with separations of ≈ h/1.42. This external particles are used to cope with the problem of kernel deficiency outside the computational domain. The wall particles are updated using the same loop as the inner fluid particles and so they are forced to satisfy Eqs. (9) and (10). However, they are not allowed to move according to Eqs. (14) and (15) so that their initial positions and velocities (v _w = 0) remain unchanged in time. In this way, the presence of the wall is modelled by means of a repulsive force, which is derived from the source term of the momentum Eq. (1) and includes the effects of compressional, viscous, and gravitational forces.

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Parallel Finite Element Method for Orographic Wind Flow and Rainfall

Kazuo Kashiyama , ... Takeo Taniguchi , in Parallel Computational Fluid Dynamics 2002, 2003

2 BASIC EQUATIONS

Assuming the viscous incompressible flow in Newtonian fluids, the wind flow can be described by the Navier-Stokes equation based on Boussinesq approximation.

(1) ${\dot{u}}_i+u_j{u}_{i,j}+\frac{1}{{\rho }_0}p,i-\nu \left(u_{i,\mathit{jj}}+u_{j,\mathit{ij}}\right)+\frac{\rho }{{\rho }_0}g{\delta }_{i3}=0$

(2) $u_{i,i}=0\mathit{in}\Omega$

(3) $\dot{\rho }+u_i\rho {,}_i=-u_i\mathit{\rho B}{,}_z{\delta }_{i3}$

where u_i is the velocity, p is the pressure, ρ is the density, ρ₀ is the reference density, V is the viscosity of fluid and δ _ij is the Kronecker's delta, respectively.

The Kessler model, which is the model for warm rain, is used for the governing equation.Figure 1 shows the schematic description of the rainfall mechanism in the Kessler model [1]. The governing equation can be given by the conservation equations for cloud and rain water:

Figure 1. Kessler model

(4) $\dot{c}+u_i{c}_{,i}+\mathit{AC}+\mathit{CC}-\mathit{EP}-\mathit{CV}=0$

(5) $\dot{r}+\left(u_i+W{\delta }_{i3}\right)r{,}_i-\mathit{AC}-\mathit{CC}+\mathit{EP}=0$

where c is the cloud water content, r is the rainwater content, u_i, is the wind velocity, W is the rainfall speed of raindrops, AC , CC, EP and CV, are the terms of auto -conversion of clouds, collection of clouds, evaporation of rain and condensation of water vapor, respectively.

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Flight (Aerodynamics)

John D. AndersonJr., in Encyclopedia of Physical Science and Technology (Third Edition), 2003

VI Inviscid Compressible Flow

Compressible flow differs from incompressible flow in at least the following respects:

1.

The density of the flow becomes a variable.

2.

The flow speeds are high enough that the flow kinetic energy becomes important and therefore energy changes in the flow must be considered. This couples the science of thermodynamics into the aerodynamic considerations.

3.

Shock waves can occur, which can completely dominate the flow.

The governing equations for inviscid, compressible flow are obtained fromEqs. (1)–(5) as

Continuity

(14) $\frac{\partial \rho }{\partial \text{t}}+\nabla \cdot \left(\rho \text{V}\right)=0$

Momentum (x component)

(15) $\rho \frac{\text{D}\text{u}}{\text{D}\text{t}}=-\frac{\partial \text{p}}{\partial \text{x}}$

Momentum (y component)

(16) $\rho \frac{\text{D}\text{v}}{\text{D}\text{t}}=-\frac{\partial \text{p}}{\partial \text{y}}$

Momentum (z component)

$\rho \frac{\text{D}\text{w}}{\text{D}\text{t}}=-\frac{\partial \text{p}}{\partial \text{z}}$

Energy

(17) $\rho \frac{\text{D}\left(\text{e}+{\text{V}}^2/2\right)}{\text{D}\text{t}}=\text{p}\dot{\text{q}}-\nabla \cdot \left(\text{p}\text{V}\right)$

Equations (14–17) are called the Euler equations. Note that for a compressible flow, Bernoulli's equation [Eq. (7) or (5)] does not hold.

The speed of sound is an important parameter for compressible flow. The speed of sound a is given by a ²  =   (∂p/∂ρ)_s, where the subscript indicates constant entropy. For a calorically perfect gas (a gas with constant specific heats and that obeys the gas law p  =   ρRT),

$\text{a}=\sqrt{\gamma \text{R}\text{T}},$

where γ   = c _p/c _v and R is the specific gas constant. In turn, the speed of sound is used to define the Mach number as

$\text{M}=\text{V}/\text{a},$

where V is the flow velocity.

Two defined properties particularly important to the analysis of compressible flow are (1) total temperature T ₀, defined as that temperature that would exist if the flow were brought to rest adiabatically, and (2) total pressure p ₀, defined as that pressure that would exist if the flow were brought to rest isentropically. For a calorically perfect gas, the relation between total and static properties is a function of γ and M only, as

$\frac{{\text{T}}_0}{\text{T}}=1+\frac{\gamma -1}{2}{\text{M}}^2$

and

(18) $\frac{{\text{p}}_0}{\text{p}}={\left(1+\frac{\gamma -1}{2}{\text{M}}^2\right)}^{\gamma /\left(\gamma -1\right)}.$

Eq. (18) is particularly useful for the measurement of Mach number in subsonic compressible flow. A Pitot tube in the flow will sense p ₀; if the static pressure p is known at the same point, then the Pitot measurement of P ₀ will directly yield the Mach number viaEq. (18).

In any steady flow where M  >   1, shock waves can occur. For example, consider the blunt-nosed, parabolically shaped body shown at the right inFig. 8. Flow is moving from left to right.If the upstream flow is supersonic (M _∞  >   1), then a curved shock wave will exist slightly upstream of the blunt nose. Two such shocks are shown to the left of the body; the leftmost shock illustrates the case for M _∞  =   4, and the rightmost shock pertains to the case for M _∞  =   8. The shock waves slow down the flow. The lower part of the flow field behind the shock wave is locally subsonic, and the upper part, after sufficient expansion around the body, is locally supersonic, albeit at a Mach number lower than M _∞. The dashed lines shown inFig. 8 divide the subsonic and supersonic regions behind the shock and are called the sonic lines, since the local Mach number M  =   1 along these lines. Note fromFig. 8 that as the freestream Mach number increases, the shock wave moves closer to the body and the sonic line moves down closer to the centerline of the flow. An essential ingredient of the understanding of supersonic flow is the calculation of the shape and strength of shock waves, such as those illustrated inFig. 8. Therefore, let us examine shock waves in more detail.

FIGURE 8. Supersonic flow over a blunt body; shock waves and sonic lines for two different freestream March numbers. [From Anderson, J. D., Jr. (2001). "Fundamentals of Aerodynamics" 3rd ed., McGraw-Hill, New York.]

A shock wave is an extremely thin region, typically on the order of 10⁻⁵ cm, across which the flow properties can change drastically. The shock wave is usually at an oblique angle to the flow, such as sketched inFig. 9a; however, there are many cases where we are interested in a shock wave normal to the flow, as sketched inFig. 9b. For example, referring again toFig. 8, the portion of the bow shock wave directly in front of the nose is normal, whereas the shock wave is oblique away from the nose. In both cases of a normal or an oblique shock, the shock wave is an almost explosive compression process, where the pressure increases almost discontinuously across the wave. As shown inFig. 9, in region 1 ahead of the shock, the Mach number, flow velocity, pressure density, temperature, entropy, total pressure, and total enthalpy (defined as e  +   ,p/ρ   + V ² /2) are denoted by M ₁, V ₁, p ₁, ρ₁, T ₁, s ₁, p _{0, 1}, and h _0,1, respectively. The analogous quantities in region 2 behind the shock are denoted by a subscript 2. The qualitative changes across the wave are noted inFig. 9. The pressure, density, temperature, and entropy increase across the shock, whereas the total pressure, Mach number, and velocity decrease. Physically, the flow across a shock wave is adiabatic, which leads to a constant total enthalpy across the wave. For a perfect gas, this also means that the total temperature is constant across the shock, i.e., T _0,1  = T _{0, 2}. In both oblique shock and normal shock cases, the flow ahead of the shock must be supersonic, that is, M ₁ > 1. Behind the oblique shock, the flow usually remains supersonic, that is, M ₂  >   1, but at a reduced Mach number, M ₂  < M ₁. However, there are a few special cases where the oblique shock is strong enough to decelerate the downstream flow to a subsonic Mach number. For the normal shock, as sketched inFig. 9b, the downstream flow is always subsonic, that is, M ₂  <   1.

FIGURE 9. Oblique and normal shock waves. [From Anderson, J. D., Jr. (2001). "Fundamentals of Aerodynamics" 3rd ed., McGraw-Hill, New York.]

The quantitative changes across a normal shock wave can be obtained fromEqs. (14)–(17) specialized for steady one-dimensional flow and integrated to obtain the following basic normal shock equations:

Continuity

${\rho }_1{\text{V}}_1={\rho }_2{\text{V}}_2$

Momentum

${\text{p}}_1+{\rho }_1{\text{V}}_1^2={\text{p}}_1+{\rho }_2{\text{V}}_2^2$

Energy

${\text{e}}_1+{\text{p}}_1/{\rho }_1+{\text{V}}_1^2/2={\text{e}}_2+{\text{p}}_2/{\rho }_2+{\text{V}}_2^2/2$

Along with the equations of state p  =   ρRT and e  =   c_v T  =   RT/(γ   −   1) for a calorically perfect gas, the basic normal shock equations can be algebraically manipulated to obtain the following changes of flow properties across the normal shock:

(19) $\frac{{\text{p}}_2}{{\text{p}}_1}=1+\frac{2\gamma }{\gamma +1}\left({\text{M}}_1^2-1\right)$

(20) $\frac{{\rho }_2}{{\rho }_1}=\frac{{\text{V}}_1}{{\text{V}}_2}=\frac{\left(\gamma +1\right){\text{M}}_1^2}{2+\left(\gamma -1\right){\text{M}}_1^2}$

(21) ${\text{M}}_2^2=\frac{1+\left[\left(\gamma -1\right)/2\right]{\text{M}}_1^2}{\gamma {\text{M}}_1^2-\left(\gamma -1\right)/2}$

Note that the changes across a normal shock wave depend only on the value of γ and the upstream Mach number M ₁.

For an oblique shock wave, let β be the angle between the shock wave and the upstream flow direction: β is called the wave angle. Also, note inFig. 9a that the flow, in crossing the oblique shock, is bent in the upward direction behind the shock. Let θ be the angle between the downstream and upstream flow directions: θ is called the deflection angle. Then,Eqs. (19)–(21) hold for an oblique shock if M ₁, V ₁, M ₂, and V ₂ are replaced by M ₁ sin β V ₁ sin β, M ₂ sin(β   −   θ), and V ₂ sin(β   −   θ), respectively. Moreover, for an oblique shock, the wave angle, deflection angle, and upstream Mach number are related through the equation

(22) $\text{tan}\theta =2\text{cot}\beta \frac{{\text{M}}_1^2{\text{sin}}^2\beta -1}{{\text{M}}_2^2\left(\gamma +\text{cos}2\beta \right)+2}.$

FromEqs. (19)–(21) written appropriately for an oblique shock, combined withEq. (22), we see that if we know any two quantities about the shock, say β and M ₁, all other quantities such as θ, M ₂, p ₂/p ₁, etc. are determined. A plot ofEq. (22) for γ   =   1.4 (air at standard conditions) is given inFig. 10. From this figure, we can deduce the following physical characteristics about oblique shock waves:

FIGURE 10. Oblique shock wave relations. [From Anderson, J. D., Jr. (1990). "Modern Compressible Flow," 2nd ed., McGraw-Hill, New York.]

1.

For any given M ₁, there is a maximum deflection angle θ_max. If the physical geometry is such that θ   >   θ_max, then no solution exists for a straight oblique shock wave; in such a case, the flow field will adjust in such a fashion to curve and detach the shock wave.

2.

For any given θ   <   θ_max, there are two values of β predicted byFig. 10 for a given M ₁. The larger value of β is called the strong shock solution, and the smaller β is the weak shock solution. In nature, the weak shock solution is favored and is the one that usually occurs.

3.

The downstream Mach number M ₂ is subsonic for the strong shock solutions, whereas M ₂ is supersonic for the weak shock solutions, except for the narrow band between the two dashed lines inFig. 10, where M ₂  <   1 even for the weak oblique shock case.

A Pitot tube can be used to measure the Mach number in a supersonic flow. However, in contrast to our previous discussions of the use of a Pitot tube, for the supersonic flow case a normal shock wave will exist in front of the mouth of the tube. Hence, the Pitot tube will measure the total pressure behind the normal shock, not the total pressure of the flow itself as in the subsonic case. By appropriate manipulation of the basic normal shock equations, the following relation can be found for a Pitot tube in supersonic flow:

(23) $\frac{{\text{p}}_{0.2}}{{\text{p}}_1}={\left[\frac{{\left(\gamma +1\right)}^2{\text{M}}_1^2}{4\gamma {\text{M}}_1^2-2\left(\gamma -1\right)}\right]}^{\gamma /\left(\gamma -1\right)}\left(\frac{1-\gamma +2\gamma {\text{M}}_1^2}{\gamma +1}\right),$

where p _0.2 is the total pressure behind the normal shock (the pressure measured by the tube), p ₁ is the static pressure upstream of the normal shock, and M ₁ is the Mach number upstream of the shock. If a Pitot tube measurement is taken at a given point in a supersonic flow, and if the static pressure is known at that point, then the Pitot measurement will directly yield the local Mach number at that point viaEq. (23).

An oblique shock, in the limit of infinitely weak strength, becomes a Mach wave, where θ   =   0, and β   =   μ, where μ is called the Mach angle:

$\mu ={\text{sin}}^{-1}\left(1/\text{M}\right).$

For all oblique shocks of a finite strength, β   >   μ.

Oblique shock waves are created when a supersonic flow is bent into itself, such as the flow over a concave corner as shown inFig. 11a. In contrast, when a supersonic flow is bent away from itself, such as the flow over a convex corner as sketched inFig. 11b, an expansion wave is created. Expansion waves are the opposite of shock waves. Expansions are composed of an infinite number of Mach waves and are a region of smooth continuous change through the expansion fan. Across the expansion wave, the Mach number increases, and the pressure, temperature, and density decrease. The flow through an expansion wave is isentropic (constant entropy), and hence, both the total pressure and the total temperature are constant through the wave. For a given Mach number M ₁ ahead of the expansion wave and a given deflection angle θ, the Mach number M ₂ behind the expansion can be found from

FIGURE 11. Oblique shock and expansion waves. [From Anderson, J. D., Jr. (2001). "Fundamentals of Aerodynamics" 3rd ed., McGraw-Hill, New York.]

$\theta =\text{v}\left({\text{M}}_2\right)-\text{v}\left({\text{M}}_1\right),$

where υ is the Prandtl–Meyer function, given by

$\text{v}\left(\text{M}\right)=\sqrt{\frac{\gamma +1}{\gamma -1}}{\text{tan}}^{-1}\sqrt{\frac{\gamma -1}{\gamma +1}\left({\text{M}}^2-1\right)}-{\text{tan}}^1\sqrt{{\text{M}}^2-1}$

For a calorically perfect gas. The corresponding temperature and pressure changes can be obtained from

$\frac{{\text{T}}_2}{{\text{T}}_1}=\frac{1+\left[\left(\gamma -1\right)/2\right]{\text{M}}_1^2}{1+\left[\left(\gamma -1\right)/2\right]{\text{M}}_2^2}$

and

$\frac{{\text{p}}_2}{{\text{p}}_1}={\left(\frac{{\text{T}}_2}{{\text{T}}_1}\right)}^{\gamma /\left(\gamma -1\right)}.$

The external supersonic flow fields over some aerodynamic bodies can sometimes be synthesized in an approximate fashion by a proper combination of local oblique shock and expansion wave solutions. Such approaches are called shock-expansion solutions.

Compressible flows through ducts (i.e., internal compressible flows) are of great importance in the design of high-speed wind tunnels, jet engines, and rocket engines, to name just a few applications. Consider a duct with a local cross-sectional area A. The area may change with length along the duct. By a proper combination of the continuity, momentum, and energy equations for a compressible flow, the following relation can be extracted, which governs the compressible flow in a variable area duct, where the flow properties are assumed to be uniform across any cross section but can change from one cross section to another (so-called quasi-one-dimensional flow):

(24) $\frac{\text{d}\text{A}}{\text{A}}=\left({\text{M}}^2-1\right)\frac{\text{d}\text{V}}{\text{V}}.$

InEq. (24), dA is the local change in area, dV is the corresponding change in velocity, and M is the local Mach number. FromEq. (24), we see that

1.

For subsonic flow (0   ≤ M  <   1), V increases as A decreases, and V decreases as A increases. Therefore, to increase the velocity, a convergent duct must be used, whereas to decrease the velocity, a divergent duct must be employed.

2.

For supersonic flow (M  >   1), V increases as A increases, and V decreases as A decreases. Hence, to increase the velocity, a divergent duct must be used, whereas to decrease the velocity, a convergent duct must be employed.

3.

When M  =   1, dA  =   0. Hence, sonic flow will occur in that location inside a variable-area duct where the area variation is a local minimum. Such a location is called a sonic throat.

Clearly, subsonic and supersonic flows through a changing-area duct behave differently. This is why subsonic wind tunnels are configured differently than supersonic wind tunnels. A sketch of a basic subsonic tunnel is shown inFig. 12. Here, we see a convergent duct for speeding up the subsonic flow (the nozzle), a constant-area section where a test model is placed in the high-speed (but subsonic) flow, and then a divergent duct (the diffuser) for slowing the flow down before exhausting it to the surroundings. In contrast, a sketch of a basic supersonic tunnel is shown inFig. 13. Here, the initially low-speed subsonic flow from a reservoir is speeded up in a convergent section, reaching Mach 1 at the throat, and then the now-supersonic flow is further speeded up in a divergent section. Hence, a nozzle designed to produce a supersonic flow is a convergent–divergent nozzle, sometimes called a de Laval nozzle, after Carl G. P. de Laval (1845–1912), who first employed such nozzles in a steam turbine. The high-speed supersonic flow then enters a constant-area test section where a test model is placed. Downstream of the model, the flow, along with the shock wave system produced by the model, enters a diffuser designed to slow the flow to a low subsonic speed before exhausting to the surroundings. To accomplish this, the initially supersonic flow must first be slowed in a convergent section to Mach 1 at a minimum area (the "second" throat) and then further slowed subsonically in a divergent section. Hence, the supersonic diffuser is also a convergent–divergent duct, just as in the case of the supersonic nozzle; however, their functions are completely opposite.

FIGURE 12. Sketch of a subsonic wind tunnel. [From Anderson, J. D., Jr. (2001). "Fundamentals of Aerodynamics" 3rd ed., McGraw-Hill, New York.]

FIGURE 13. Sketch of a supersonic wind tunnel. [From Anderson, J. D., Jr. (2001). "Fundamentals of Aerodynamics" 3rd ed., McGraw-Hill, New York.]

The flow in a properly designed supersonic nozzle is isentropic. For this case, and for a calorically perfect gas, the Mach number that exists at a given cross section with area A is simply a function of the ratio of this area to the throat area, denoted by A*. That is,

${\left(\frac{\text{A}}{{\text{A}}^{*}}\right)}^2=\frac{1}{{\text{M}}^2}{\left[\frac{2}{\gamma +1}\left(1+\frac{\gamma -1}{2}{\text{M}}^2\right)\right]}^{\left(\gamma +1\right)/\left(\gamma -1\right)}$

This is a double-valued function; for a given A/A*, two solutions exist for M—a subsonic value in the convergent portion of the nozzle and a supersonic value in the divergent portion.

Let us return to the consideration of external compressible flows. If the flow is subsonic or supersonic, irrotational, and involves the flow over slender bodies at small angle of attack,Eqs. (14)–(17) reduce to a single linearized equation:

(25) $\left(1-{\text{M}}_{\infty }^2\right)\frac{{\partial }^2\varphi }{\partial {\text{x}}^2}+\frac{{\partial }^2\varphi }{\partial {\text{y}}^2}+\frac{{\partial }^2\varphi }{\partial {\text{z}}^2}=0,$

where ϕ is a perturbation velocity potential defined such that ∂ϕ/∂x= u, ∂ϕ/∂ y  =   υ, and ∂ϕ/∂ z  = w, where u, υ, and w are small perturbation velocities superimposed on the uniform free stream in such a manner that u  = V _∞  + u, υ   =   υ, and w  = w.Equation (25) does not hold for transonic or hypersonic Mach numbers.

A solution ofEq. (25) for a subsonic flow leads to a method for correcting low-speed, incompressible flow data to take into account compressibility effects—so-called compressibility corrections. For example, if C _po denotes the low-speed incompressible flow value of the pressure coefficient at a certain point on a two-dimensional airfoil, then the value of C _p at the same point for a high-speed subsonic freestream Mach number M _∞ is given approximately by ${\text{C}}_{\text{p}}={\text{C}}_{\text{p}0}/\sqrt{1-{\text{M}}_{\infty }^2}$ . This compressibility correction is called the Prandtl–Glauert rule and is generally valid for M _∞  <   0.7. In turn, the lift and moment coefficients for the airfoil are similarly related to their corresponding low-speed incompressible values ${\text{c}}_{{\text{l}}_0}$ and ${\text{c}}_{{\text{m}}_0}$ as

${\text{c}}_{\text{l}}=\frac{{\text{c}}_{{\text{l}}_0}}{\sqrt{1-{\text{M}}_{\infty }^2}}{\text{c}}_{\text{m}}=\frac{{\text{c}}_{{\text{m}}_0}}{\sqrt{1-{\text{M}}_{\infty }^2}}.$

Equation (25) can also be solved for supersonic flows, leading to a simple expression for the pressure coefficient for two-dimensional flow:

(26) ${\text{C}}_{\text{p}}=\frac{2\theta }{\sqrt{{\text{M}}_{\infty }^2-1}},$

where θ is the local angle between the tangent to the body and the freestream direction and M _∞ is the freestream Mach number. In turn, this result from linearized theory can be used to calculate the lift and wave drag coefficients for thin airfoils at a small angle of attack. For example, for a flat plate at angle of attack α,Eq. (26) leads to

(27) ${\text{c}}_{\text{l}}=\frac{4\alpha }{\sqrt{{\text{M}}_{\infty }^2-1}},$

(28) ${\text{c}}_{\text{d}}=\frac{4{\alpha }^2}{\sqrt{{\text{M}}_{\infty }^2-1}}.$

The wave drag coefficient c _d is a ramification of the high pressure behind a shock wave and can be associated with the entropy increase across a shock wave. We can view wave drag as that component of drag that is associated with shock waves on the body; obviously, wave drag exists only at supersonic and hypersonic speeds. For thin airfoils at small α,Eq. (27) still holds; however,Eq. (28) is modified to take into account the finite thickness and curvature (camber) of the airfoil.

Modern exact calculations of inviscid compressible flows—subsonic, transonic, supersonic, and hypersonic—are made by means of sophisticated numerical solutions ofEqs. (14)–(17) on a high-speed digital computer.

As a final note in this section, a comment is made about hypersonic flow. Such flow, which by rule of thumb is denoted as the regime where M _∞  >   5, is characterized by several important physical phenomena already described inSection IV. Among these is the existence of thin shock layers around the body (the shock waves lie close to the body). For such a flow, the physical flow picture looks very much like the original model proposed by Newton in 1687 (as described inSection I). Hence, Newton's sine-squared law is a reasonable prediction for the pressure coefficient at hypersonic speeds, that is,

${\text{C}}_{\text{p}}=2{\text{sin}}^2\theta ,$

where θ is the angle between a tangent to the surface and the freestream direction. A modified Newtonian formula actually gives improved results; that is,

${\text{C}}_{\text{p}}={\text{C}}_{{\text{p}}_{\text{max}}}{\text{sin}}^2\theta ,$

where ${\text{C}}_{\text{pmax}}$ is the pressure coefficient at the stagnation point after the flow has passed through a normal shock wave at the freestream Mach number.

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A Parallel Solenoidal Basis Method for Incompressible Fluid Flow Problems*

Sreekanth R. Sambavaram , Vivek Sarin , in Parallel Computational Fluid Dynamics 2001, 2002

1 INTRODUCTION

Large-scale simulation of incompressible flow is one of the most challenging application. Realistic simulations are possible only with the use of sophisticated modeling techniques, preconditioned iterative methods and advanced parallel architectures. The motivation for this work is to develop an effective approach for solving the linear systems arising in incompressible flows with high efficiency on a multi-processor platform.

The principles of classical mechanics, thermodynamics, and laws of conservation of mass, momentum, and energy govern the motion of the fluid. Law of conservation of momentum for incompressible, viscous flow in a region Ω with boundary ∂Ω is captured by Navier-Stokes equation given by

(1) $\frac{\partial \mathbf{u}}{\partial t}+\mathbf{u}\cdot \nabla \mathbf{u}=‐\nabla p+\frac{1}{R}\mathrm{\Delta }\mathbf{u}\text{,}$

where p  = p(x, t) is the pressure, R is the Reynolds number, and u  = u(x, t) is the velocity vector at x. The law of conservation of mass for incompressible fluids gives rise to the continuity equation

(2) $\nabla \cdot \mathrm{u}=0\mathrm{in}\Omega \text{.}$

Appropriate boundary conditions may be specified for fluid velocity. Suitable discretization and linearization of the equations(1)–(2) result in the following linear system

(3) $\left[\begin{array}{cc}\hfill A\hfill & \hfill B\hfill \\ \hfill B^T\hfill & \hfill 0\hfill \end{array}\right]\left[\begin{array}{c}\hfill u\hfill \\ \hfill p\hfill \end{array}\right]=\left[\begin{array}{c}\hfill f\hfill \\ \hfill 0\hfill \end{array}\right]\text{,}$

where B^T is the discrete divergence operator and A is given by

(4) $A=\frac{1}{\mathit{\Delta t}}M+C+\frac{1}{R}L\text{,}$

in which M is the mass matrix, L is the Laplace matrix, and C is the matrix arising from the convection term. When operator splitting is used to separate the linear and non-linear terms, we obtain the generalized Stokes problem (GSP) with a symmetric positive definite A given as

(5) $A=\frac{1}{\mathit{\Delta t}}M+\frac{1}{R}L\text{.}$

The linear system (3) is large and sparse. Although direct methods can be used to solve this system, they require prohibitively large amount of memory and computational power. The inherent sequential nature of these techniques limits the efficiency on parallel architectures. In contrast, iterative methods require significantly less memory and are well suited for parallel processing. These methods can be made more reliable by using preconditioning techniques which accelerate convergence to the solution. In order to make iterative methods more competitive, one must devise robust preconditioning techniques that are not only effective but parallelizable as well.

This paper presents a preconditioned solenoidal basis method to solve the linear system (3) arising in the generalized Stokes problem. Section 2 describes the solenoidal basis method and section 3 outlines the preconditioning scheme. Experiments for the driven cavity problem in 2D and 3D are presented in section 4.

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Wavelet Zoom

Stéphane Mallat , in A Wavelet Tour of Signal Processing (Third Edition), 2009

Hydrodynamic Turbulence

Fully developed turbulence appears in incompressible flows at high Reynolds numbers. Understanding the properties of hydrodynamic turbulence is a major problem of modern physics, which remains mostly open despite an intense research effort since the first theory of Kolmogorov in 1941 [ 331]. The number of degrees of liberty of three-dimensional turbulence is considerable, which produces extremely complex spatio-temporal behavior. No formalism is yet able to build a statistical physics framework based on the Navier-Stokes equations that would enable us to understand the global behavior of turbulent flows as it is done in thermodynamics.

In 1941, Kolmogorov [331] formulated a statistical theory of turbulence. The velocity field is modeled as a process V(x) that has increments with variance

$E\left\{\right|V(x+\Delta )-V\left(x\right){|}^2\}\sim {\varepsilon }^{2/3}{\Delta }^{2/3}\mathrm{.}$

The constant ɛ is a rate of dissipation of energy per unit of mass and time, which is supposed to be independent of the location. This indicates that the velocity field is statistically homogeneous with Lipschitz regularity a = H = 1/3. The theory predicts that a one-dimensional trace of a three-dimensional velocity field is a fractal noise process with stationary increments that have spectrum decays with a power exponent 2H + 1 = 5/3:

${\hat{R}}_V\left(\omega \right)=\frac{{\sigma }_H^2}{{\left|\omega \right|}^{5/3}}\mathrm{.}$

The success of this theory comes from numerous experimental verifications of this power spectrum decay. However, the theory does not take into account the existence of coherent structures such as vortices. These phenomena contradict the hypothesis of homogeneity, which is at the root of Kolmogorov's 1941 theory.

Kolmogorov [332] modified the homogeneity assumption in 1962 by introducing an energy dissipation rate ɛ(x) that varies with the spatial location x. This opens the door to "local stochastic self-similar" multifractal models, first developed by Mandelbrot [370] to explain energy exchanges between fine-scale structures and large-scale structures. The spectrum of singularity D(α) is playing an important role in testing these models [264]. Calculations with wavelet maxima on turbulent velocity fields [5] show that D(α) is maximum at 1/3, as predicted by the Kolmogorov theory. However, D(α) does not have a support reduced to {1/3}, which verifies that a turbulent velocity field is not a homogeneous process. Models based on the wavelet transform have been introduced to explain the distribution of vortices in turbulent fluids [13, 251, 252].

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Parallel Efficiency of a Variable Order Method of Lines

Hidetoshi Nishida , ... Nobuyuki Satofuka , in Parallel Computational Fluid Dynamics 2002, 2003

2.1 Governing Equations

In this paper, we consider the incompressible flows. Then, the incompressible Navier-Stokes equations can be written by

(1) $\frac{\partial u_i}{\partial x_i}=0\text{,}$

(2) $\frac{\partial u_i}{\partial t}+u_j\frac{\partial u_i}{\partial x_j}=–\frac{\partial p}{\partial x_i}+\frac{1}{\mathit{Re}}\frac{{\partial }^2{u}_i}{\partial x_j\partial x_j}\text{.}$

Equation (1) denotes the continuity equation and Eq. (2) the momentum equations. The velocity components and the pressure are expressed by u_i and p, respectively. Re denotes the Reynolds number defined by Re = UL/ν, where v is the kinematic viscosity. The equations are nondimensionalized by the reference length L, the reference velocity U and the reference pressure ρU ².

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