Download Computational Hydraulics Equation, Lecture Notes- Physics - and more Study notes Physics in PDF only on Docsity!
3. APPROXIMATIONS AND SIMPLIFIED EQUATIONS SPRING 2012
3.1 Steady-state vs time-dependent flow 3.2 Two-dimensional vs three-dimensional flow 3.3 Incompressible vs compressible flow 3.4 Inviscid vs viscous flow 3.5 Hydrostatic vs non-hydrostatic flow 3.6 Boussinesq approximation for density (buoyancy-affected flow) 3.7 Depth-averaged / shallow-water equations 3.8 Reynolds-averaged equations (turbulent flow) Examples
Fluid dynamics is governed by equations for mass, momentum and energy. The momentum equation for a viscous fluid is called the Navier-Stokes equation; it is based upon:
- continuum mechanics;
- the momentum principle;
- shear stress proportional to velocity gradient.
A fluid for which the last is true is called a Newtonian fluid; this is the case for almost all fluids in civil engineering. However, there are important non-Newtonian fluids; e.g. mud, blood, paint, polymer solutions, volcanic lava. Specialised CFD techniques exist for these.
The full equations are time-dependent, 3-dimensional, viscous, compressible, non-linear and highly coupled (i.e. the solution of one equation depends on the solution of others). However, in most cases it is possible to simplify the analysis by adopting a reduced equation set. Some common approximations are listed below.
Reduction of dimension :
- steady-state;
- 2-d (or axisymmetric).
Neglect of some physical feature :
- incompressible;
- inviscid.
Simplified forces :
- hydrostatic;
- Boussinesq density.
Approximations based upon averaging :
- Reynolds-averaging (turbulent flows);
- depth-averaging (shallow-water equations).
3.1 Steady-State vs Time-Dependent Flow
Many flows are naturally time-dependent. Examples include waves, tides, reciprocating pumps and internal combustion engines.
Other flows have stationary boundaries but become time-dependent because of an instability. An important example is vortex shedding around cylindrical objects.
Some computational solution procedures rely on a time-stepping method to march to steady state ; examples are transonic flow and open-channel flow (where the mathematical nature of the governing equations is different for Mach or Froude numbers less than or greater than 1).
Thus, there are three major reasons for using the full time-dependent equations:
- time-dependent problem;
- time-dependent instability;
- time-marching to steady state.
Numerical Consequences.
- Time-dependent equations are parabolic (in incompressible flow); they are 1st-order in time and solved by time-marching , only storing values at one or two time levels.
- Steady-state equations are elliptic (in incompressible flow) and are solved by implicit, iterative methods for all points simultaneously.
3.2 Two- Dimensional vs Three-Dimensional Flow
Geometry and boundary conditions may dictate that the flow is two-dimensional (at least in the mean). Two-dimensional calculations require considerably less computer resources.
“Two-dimensional” may be extended to include “axisymmetric”. This is actually easier to achieve in the laboratory, where side-wall boundary layers prevent true 2-dimensionality.
3.3 Incompressible vs Compressible Flow
Compressibility is important when flow-induced variations in pressure or temperature cause significant changes in density. This can arise: (a) in high-speed flow; (b) where there is significant heat input.
A flow (not a fluid, note) is said to be incompressible if flow-induced pressure or temperature changes do not cause significant density changes. Liquid flows are usually treated as incompressible (water hammer being an important exception), but gas flows can also be regarded as incompressible at speeds much less than the speed of sound (a common rule of thumb being Ma < 0.3).
Density variations within the fluid can occur for other reasons, notably from salinity variations (oceans) and temperature variations (atmosphere). These lead to buoyancy forces. Because the density variations are not flow-induced these flows can still be treated as incompressible, even though the density varies from place to place. Thus, “incompressible” does not necessarily mean “uniform density”.
3.4 Inviscid vs Viscous Flow
If viscosity is neglected, the Navier-Stokes equations become the Euler equations.
Consider streamwise momentum in a developing 2-d boundary layer:
2
2 ( ) y
u x
p y
u v x
u u ∂
mass × acceleration = pressure force + viscous force
Dropping the viscous term reduces the order of the highest derivative from 2 to 1 and hence one less boundary condition is required.
- Viscous (real) flows require a no-slip condition (zero velocity) at rigid walls – the dynamic boundary condition.
- Inviscid (ideal) flows require only the velocity component normal to the wall to be zero – the kinematic boundary condition. The wall shear stress is zero.
Although its magnitude is small, and consequently its direct influence via the shear stress is tiny, viscosity can have a global influence out of all proportion to its size. The most important example is flow separation , where the viscous boundary layer required to satisfy the non-slip condition is first slowed and then reversed by an adverse pressure gradient. Boundary- layer separation has two important consequences:
- major disturbance to the flow;
- a large increase in pressure drag.
3.4.1 Inviscid Approximation: Potential Flow
Velocity Potential, φ
In inviscid flow it may be shown^1 that the velocity components can be written as the gradient of a single scalar variable, the velocity potential φ:
x
u ∂
∂ φ = , y
v ∂
∂ φ = , z
w ∂
∂ φ = (also written: u =∇φ)
Substituting these into the continuity equation for incompressible flow:
= 0 ∂
z
w y
v x
u
gives
(^1) Since pressure acts perpendicularly to a surface and cannot impart rotation an inviscid fluid can be regarded as
irrotational ( ∇ ∧ u = 0 ), and so the velocity field can be written as the gradient of a scalar function.
y U
y U
viscous
inviscid
recirculating flow
separation
2 2
2 2
2
∂
∂ φ
∂
∂φ
∂
∂φ x y z
which is often abbreviated as
∇^2 φ^ = 0 (Laplace’s equation) (1)
Stream Function,
In 2-d incompressible flow we will see in Section 9 that there exists a function called the stream function such that
y
u ∂
x
v ∂
If the flow is also inviscid then it may be shown that the fluid is irrotational and
= 0 ∂
x
v y
u
Substituting the expressions for u and v into this gives an equation for :
∇^2 = 0 (2)
In both cases above the entire 3-d flow field is completely described by a single scalar field, either φ or. Moreover, since Laplace’s equation occurs in many branches of physics (electrostatics, heat conduction, gravitation, optics, ...) many good solvers exist.
Velocity components u , v and w are obtained by differentiating φ or. Pressure is then recoverable from Bernoulli’s equation:
p + 21 U^2 = constant (along a streamline)
where U is the magnitude of velocity.
The potential-flow assumption often gives an adequate description of the flow and pressure fields for real fluids, except very close to solid surfaces where viscous forces are significant. It is useful, for example, in calculating the lift force on aerofoils and in wave theory. However, in ignoring the effects of viscosity it implies that there are neither tangential stresses on boundaries nor flow separation, which leads to the erroneous conclusion that an object moving through a fluid experiences no drag.
3.4.2 Potential Flow + Boundary Layer
For non-separated flows it is sometimes acceptable to approximate the flow domain as an inviscid outer region (where the flow is driven by pressure gradients arising from the shape of the boundary), matched to a thin inner layer (across which the pressure is effectively constant) to satisfy the no-slip condition. The inner-layer solution is often obtained by forward-marching in space ( parabolised Navier-Stokes equations) in a manner similar to time-marching.
This “viscid-inviscid interaction” technique is widely used for aerofoils: outer flow → pressure distribution and lift; inner flow → viscous drag (usually small).
boundary layer
outer flow
3.6 Boussinesq Approximation for Density (Buoyancy-Affected Flow) 2
For constant-density flows, pressure and gravitational forces in the vertical momentum equation can be combined through the piezometric pressure p *^ = p + gz. Gravity need not explicitly enter the flow equations unless:
- pressure enters the boundary conditions (e.g. at a free surface, where p = patm );
- there is variable density. Density variations may arise even at low speeds because of changes in temperature or humidity (atmosphere), or salinity (water).
Temperature variations in the atmosphere, brought about by surface (or, occasionally, cloud-top) heating or cooling, are responsible for significant changes in airflow and turbulence.
- On a cold night the atmosphere is stable. Cool, dense air collects near the surface and vertical motions are suppressed; the boundary layer depth is 100 m or less.
- On a warm day the atmosphere is unstable. Surface heating produces warm; light air near the ground and convection occurs; the boundary layer may be 2 km deep. The vertical dispersion of pollution is very different in the two cases.
If density is a function of some scalar (typically, temperature or, in the oceans, salinity), then the relative change in density is proportional to the change in ; i.e.
0
= or − 0 = 0 ( − 0 )
where 0 and 0 are reference scalar and density and is the coefficient of expansion; (the sign convention adopted here is that for salinity, where an increase in salinity leads to a rise in density – the opposite would be true for temperature-driven density changes).
The Boussinesq approximation for density amounts to retaining density variations in the gravitational forces but disregarding them in the inertial (mass × acceleration) term; i.e.
g z
p t
w D
D
or, in terms of θ, with the part of the weight resulting from the constant reference density (^0) subsumed into a modified pressure p *:
142 43 buoyancyforce
g z
p t
w * D
D
= − where p * = p + 0 gz
The approximation is justified if relative density variations are not too large; i.e.
1 0
«
This condition is usually satisfied in the atmosphere and oceans.
(^2) Note that several other very-different approximations are also referred to as the Boussinesq approximation in
different contexts – e.g. shallow-water equations or eddy-viscosity turbulence models.
ρ u
ρ (^) u
stable boundary layer
convective boundary layer
mixing depth
Actually, whilst the Boussinesq approximation may be necessary to obtain analytical solutions, it is not particularly important in general-purpose CFD because the momentum and scalar transport equations are solved iteratively and any scalar-dependent density variations are easily incorporated in the iterative procedure.
3.7 Depth-Averaged / Shallow-Water Equations
This approximation is used for flow of a constant-density fluid with a free surface, where the depth of fluid is small compared with typical horizontal scales.
In this “hydraulic” approximation, the fluid can be regarded as quasi-2d with:
- horizontal velocity components u , v ;
- depth of water, h.
Note that h may vary due to changes in both free- surface and bed height.
By applying mass and momentum principles to a vertical column of constant-density fluid of variable height h , the depth-integrated equations governing the motion can be written (for the one-dimensional case and in conservative form) as:
2 2
(^21)
x surface^ bed
gh x
u h t
uh
x
uh t
h
The 21 gh^2 term comes from (1/ times) the hydrostatic pressure force per unit width on a
water column of height h ; i.e. average hydrostatic pressure ( 12 gh ) × height ( h ). The final
term is the net effect of surface stress (due to wind) and the bed shear stress (due to friction). These equations are derived in the Examples.
The resulting shallow-water (or Saint-Venant ) equations are mathematically similar to those for a compressible gas. There are direct analogies between
- hydraulic jumps (shallow flow) and shocks (compressible flow);
- critical flow over a weir (shallow) or gas flow through a throat (compressible).
In both cases there is a characteristic wave speed ( c = gh in the hydraulic case, c = p /
in compressible flow). Depending on whether this is greater or smaller than the flow velocity determines whether disturbances can propagate upstream and hence the nature of the flow. The ratio of fluid speed to wave speed is known as:
Froude number : gh
u Fr = in shallow flow
Mach number : c
u Ma = in compressible flow
h(x,t)
z
x
u
Examples
Q1. Discuss the circumstances under which a fluid flow can be approximated as: (a) incompressible; (b) inviscid.
Q2.
By resolving forces along a streamline, the steady-state momentum equation for an inviscid fluid can be written
g sin s
p s
U
U −
where U is the velocity magnitude, s the distance along a streamline and the angle between local velocity and the horizontal. Assuming incompressible flow derive Bernoulli’s equation. (This question demonstrates that, for incompressible flow, the mechanical energy principle can be derived directly from the momentum equation.)
Q3.
A velocity field is given by the velocity potential φ = x^2 − y^2.
(a) Calculate the velocity components u and v. (b) Calculate the acceleration. (c) Calculate the corresponding streamfunction,. (d) Sketch the streamlines and suggest a geometry in which one might expect this flow.
Q4.
For incompressible flow in a rotating reference frame the force per unit volume f is the sum of pressure, gravitational, Coriolis and viscous forces:
f = −∇ p + g − 2 ∧ u + ∇^2 u
where g = (0, 0, – g ) is gravity and is the angular velocity of the rotating frame.
(a) If the density is uniform, show that pressure and gravitational forces can be combined in a piezometric pressure (which should be defined).
(b) If density variations arise from a scalar field, describe the “Boussinesq” approximation (in this context) and give an application in which it is used.
(c) Show how the momentum equation (with Boussinesq approximation for density) can be non-dimensionalised in terms of (densimetric) Froude number, Rossby number and Reynolds number:
, Ro , Re ( / )
Fr 0 0 0 0
0 0 0
0 UL
L
U
gL
U
where 0 , L 0 , U 0 are characteristic density, length and velocity scales, respectively, and is a typical magnitude of density variations.
α
U
s
Q5.
(a) In flow with a free surface, by taking a control volume as a column of (time-varying) height h ( x , y , t ) and horizontal cross-section x × y , assuming that density is constant, 13 and^23 are the only significant stress components, the horizontal velocity field may be replaced by the depth-averaged velocity ( u , v ) and the pressure is hydrostatic, derive the shallow-water equations for continuity and x -momentum in the form
0
y
hv x
hu t
h
( ) (^2 ) ( ) 13 ( surface ) 13 ( bed ) x
z gh y
hvu x
hu t
hu (^) s −
∂
(b) Provide an alternative derivation by integrating the continuity and horizontal momentum equations for incompressible flow:
= 0 ∂
z
w y
v x
u
x z
p z
wu y
vu x
u t
u ∂
∂ ( ) (^2 ) ( ) ( ) 13
over a depth h = zs – zb.
For part (b) you will need the boundary condition that the top and bottom surfaces z = zs ( x , y )and z = zb ( x , y )are material surfaces:
( ) 0 D
D
z − zs = t
or = 0 ∂
y
z v x
z u t
z w s^ s s on z = zs
and similarly for zb , together with Leibniz’ Theorem for differentiating an integral:
x
a f a x
b x f b x
f f x x x
b x
ax
bx
a x d
d ( ) d
d ( ) d d ( ) d
d ()
()
()
()
+^ −
Note: this is easily extended to consider additional forces such as Coriolis forces and other stress terms (“horizontal diffusion”). For undergraduates Dr Rogers will cover this in the second part of the Computational Hydraulics course.
