Studies in CFD – Summer 2012 - Homework 1
1.- Consider a volume element inside a flowing fluid and write down an
expression representing the mass conservation principle inside the volume (i.e.
rate of mass in = rate of mass out + rate of mass accumulation). Then, use Gauss’
theorem to convert the surface integral representing the net mass inflow into a
volume integral and obtain the continuity equation.
The conservation law relates the rate of change of an amount to that property in a
given control mass. For mass, the conservation equation can be written:
𝑑𝑚
=0
𝑑𝑡
If φ is a conserved intensive property; i.e. mass conservation φ = 1, then the
corresponding extensive property Φ can be expressed as:
Φ = ∫𝛺𝐶𝑀 𝜌𝜙𝑑𝛺
where ΩCM stands for the volume occupied by the CM. Using this definition, the left
hand side of each conservation equation for a control volume can be written as:
𝑑
𝑑
∫ 𝜌𝜙𝑑𝛺 =
∫ 𝜌𝜙𝑑𝛺 + ∫ 𝜌𝜙(𝑣 − 𝑣𝑏 ) · 𝑛𝑑𝑆
𝑑𝑡 𝛺𝐶𝑀
𝑑𝑡 𝛺𝐶𝑉
𝑆𝐶𝑉
ΩCV = Control Volume
SCV = surface enclosing the Control Volume
n is the unit vector orthogonal to the SCV
v is the fluid velocity and vb is the velocity with which the CV surface is
moving.
Since mass conservation was defined as φ = 1:
𝜕𝑦
∫ 𝜌𝑑𝛺 + ∫ 𝜌𝑣 · 𝑛𝑑𝑆
𝜕𝑥 𝛺
𝑆
0
By applying Gauss’=
divergence
theorem to the convection term, transforming a
surface integral to a volume integral and setting the volume infinitesimally small:
𝜕𝜌
+ 𝑑𝑖𝑣(𝜌𝑣 )
𝜕𝑡
=0
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Using this form in Cartesian form:
𝜕(𝜌𝑢𝑦 )
𝜕𝜌
𝜕(𝜌𝑢𝑖 )
𝜕𝜌
𝜕(𝜌𝑢𝑥 )
𝜕(𝜌𝑢𝑧 )
+
=
+
+
+
𝜕𝑡
𝜕𝑢𝑖
𝜕𝑡
𝜕𝑥
𝜕𝑦
𝜕𝑧
=0
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2.- The stress tensor in a Newtonian fluid, representing the molecular transport of
momentum consists of two parts; explain what is represented in each part and
the difference(s) between them.
For Newtonian fluids, the Stress tensor, T, which is the molecular rate of transport
of the momentum, can be written as:
T = -(𝑝 +
2
3
𝜇 𝑑𝑖𝑣 𝑣) I + 2μD
where μ is the dynamic viscosity
I is the unit tensor
p is the static pressure
D is the strain rate tensor
Note:
1
𝐷 = 2 [𝑔𝑟𝑎𝑑 𝑣 + (𝑔𝑟𝑎𝑑 𝑣)𝑇 ]
The two equations can be written in Cartesian coordinates as
2 𝜕𝑢𝑗
𝑇𝑖𝑗 = − (𝑝 + 𝜇
)δ
3 𝜕𝑥𝑗 𝑖𝑗
2𝜇𝐷
1 +𝜕𝑢
𝑖 𝑖𝑗𝜕𝑢𝑗
𝐷𝑖𝑗 = (
+
)
2 𝜕𝑥𝑗 𝜕𝑥𝑖
where δij is the Kronecker symbol(δij = 1 if i = j and δij = 0 otherwise). For
incompressible flows, the second term in equation
2 𝜕𝑢𝑗
𝜇
)δ
3 𝜕𝑥𝑗 𝑖𝑗
is zero by virtue of the continuity equation.
+ 2𝜇𝐷𝑖𝑗
𝑇𝑖𝑗 = − (𝑝 +
I need more here to further describe the 2
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3.- Consider the slow, steady state laminar flow of a Newtonian fluid contained
between two large, horizontal parallel plates driven by a constant pressure
gradient dp/dx. Consider a small volume element of fluid of thickness dz in the
vertical direction and length dx in the direction of the flow and write down an
expression representing the momentum conservation principle in the volume.
Finally, make the volume of the element go to zero and derive the momentum
conservation equation for the system.
𝑂𝑢𝑡𝑓𝑙𝑜𝑤 𝑜𝑓 𝑚𝑜𝑚𝑒𝑛𝑡𝑢𝑚 = [𝐺𝑥 𝑢 +
𝜕
𝜕𝑥
(𝐺𝑥 𝑢)𝛿𝑥]δy + [𝐺𝑦 𝑢 +
𝜕
𝜕𝑦
(𝐺𝑦 𝑢)𝛿𝑦]δx
𝑖𝑛𝑓𝑙𝑜𝑤 𝑜𝑓 𝑚𝑜𝑚𝑒𝑛𝑡𝑢𝑚 = (𝐺𝑥 𝛿𝑦)𝑢 + (𝐺𝑦 𝛿𝑥)𝑢
Increase of momentum storage – 0 (steady flow)
𝐸𝑥𝑡𝑒𝑟𝑛𝑎𝑙 𝑓𝑜𝑟𝑐𝑒𝑠 = −𝜎𝑥 𝛿𝑦 − 𝜏𝑦𝑥 𝛿𝑥 + (𝜎𝑥 +
𝜕𝜎𝑥
𝜕𝑥
𝛿𝑥)δy + (𝜏𝑦𝑥 +
𝜕𝜏𝑦𝑥
𝜕𝑦
𝛿𝑦)δx
Applying the momentum theorem with the assumption of steady flow and expanding
terms yields
𝜕𝐺𝑦
𝜕𝜏𝑦𝑥
𝜕𝑢
𝜕𝐺𝑥
𝜕𝑢
𝜕𝜎𝑥
𝐺𝑥 the
+ 𝑢continuity
+ 𝐺equation
+ u using=the assumption
+
Applying
𝑦
𝜕𝑥
𝜕𝑥
𝜕𝑥
𝜕𝑥
𝜕𝑥
𝜕𝑦 of steady flow yields
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𝜕𝜏𝑦𝑥
𝜕𝑢
𝜕𝑢
𝜕𝜎𝑥
+ 𝐺𝑦
=
+
𝜕𝑥
𝜕𝑥
𝜕𝑥
𝜕𝑦
Then, applying the boundary-layer approximation to the shear stress term and the
normal stress term gives us
𝐺𝑥
𝜕𝜎𝑥
𝑑𝑃
=−
𝜕𝑥
𝑑𝑥
𝜕𝜏𝑦𝑥
𝜕
𝜕𝑢
=
(𝜇 )
𝜕𝑦
𝜕𝑦 𝜕𝑦
Note that:
Gx = ρu, Gy = ρv
With these substitutions, we have the momentum equation of the boundary layer
𝜕𝑢
𝜕𝑢
𝑑𝑃
𝜕
𝜕𝑢
+ 𝜌𝑣
=−
+
(𝜇 )
𝜕𝑥
𝜕𝑦
𝑑𝑥
𝜕𝑦
𝜕𝑦
Note that this equation is valid for variable properties ρ and μ.
𝜌𝑢
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4.- Consider a flow in a 3D rectangular Cartesian coordinate system and write
down all the terms represented by the operator expression (v · ∇) that appears in
the inertia term of the non-conservative form of the momentum equation (Eqn
1.20 in text).
𝛻·𝑉 =
Michael Wegrzyniak
𝜕𝑉𝑦
𝜕𝑉𝑥
𝜕𝑉𝑧
+
+
𝜕𝑥
𝜕𝑦
𝜕𝑧
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5.- Consider a flow in a 3D cylindrical polar system of coordinates and write
down the component form expressions of the continuity and momentum
conservation equations for the flow.
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6.- Dimensionless numbers are always ratios of two distinct quantities, both with
the same physical units. Describe the two quantities involved in the following
dimensionless numbers used in CFD: St, Re, Fr and Ra.
Stanton Number:
St is the ratio of heat transfer to thermal capacity.
ℎ
St = 𝜌𝐶
𝑝𝑉
Reynolds Number:
Re is the ratio of inertial force to viscous force.
𝑅𝑎 =
𝜌𝑉𝐿
𝜇
Froude Number:
FR is the ratio of inertial force to gravitational force.
𝑉2
𝐹𝑟 =
𝑔𝐿
Rayleigh Number:
Ra is the ration of buoyancy force to viscous force
𝑔𝛽|𝛥𝑇|𝐿3 𝜌2 𝐶𝑝
𝑅𝑎 =
𝑘𝜇
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7.- Explain briefly the assumptions involved in creating the following
simplified models:
a) Incompressible Flow
In many applications the fluid density may be assumed constant. If the Mach
number is below 0.3 compressibility may indeed be neglected If the flow is also
isothermal, the viscosity is also constant.
b) Inviscid Flow
In flows far from solid surfaces, the effects of viscosity are usually very small. If
viscous effects are neglected altogether, i.e. if we assume that the stress tensor
reduces to T = -pl, the Navier-Stokcs equations reduce to the Euler equations.
c) Potential Flow
The fluid is assumed to be inviscid (as in the Euler equations); however, an
additional condition is imposed on the flow - the velocity field must be irrotational.
d) Creeping Flow
When the flow velocity is very small, the fluid is very viscous, or the geometric
dimensions are very small, the convective (inertial) terms in the Navier-Stokes
equations play a minor role and can be neglected. The flow is then dominated by
the viscous, pressure, and body forces and is called creeping flow. If the fluid
properties can be considered constant, the momentum equations become linear;
they are usually called Stokes equations. Due to the low velocities the unsteady
term can also be neglected, a substantial
e) Boussinesq Approximation
In flows accompanied by heat transfer, the fluid properties are normally functions
of temperature. The variations may be small and yet be the cause of the fluid
motion. If the density variation is not large, one may treat the density as constant in
the unsteady and convection terms, and treat it as variable only in the gravitational
term.
f) Boundary Layer Approximation
When the flow has a predominant direction and the variation of the geometry is
gradual, the flow is mainly influenced by what happened upstream. Examples are
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flows in channels and pipes and flows over plane or mildly curved solid walls.
Such flows are called thin shear layer or boundary layer flows. The Navier-Stokes
equations can be simplified for such flows as follows:
 diffusive transport of momentum in the principal flow direction is much
smaller than convection and can be neglected;
 the velocity component in the main flow direction is much larger than the
components in other directions;
 the pressure gradient across the flow is much smaller than in the principal
flow direction.
The simplified equations can be solved by using marching techniques similar to
those used to solve ordinary differential equations with initial conditions. These
techniques see considerable use in aerodynamics.
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Studies in CFD – Summer 2012 - Homework 1
8.- Explain briefly the key characteristics of the following mathematical types
of flows:
a) Hyperbolic Flows
In the hyperbolic case, the characteristics are real and distinct. This means that
information propagates at finite speeds in two sets of directions. In general, the
information propagation is in a particular direction so that one datum needs to be
given at an initial point on each characteristic; the two sets of characteristics
therefore demand two initial conditions. If there are lateral boundaries, usually only
one condition is required at each point because one characteristic is carrying
information out of the domain and one is carrying information in.
b) Parabolic Flows
In parabolic equations the characteristics degenerate to a single real set.
Consequently, only one initial condition is normally required. At lateral boundaries
one condition is needed at each point.
c) Elliptic Flows
In the elliptic case, the characteristics are imaginary or complex so there are no
special directions of information propagation. Indeed, information travels
essentially equally well in all directions. Generally, one boundary condition is
required at each point on the boundary and the domain of solution is usually closed
although part of the domain may extend to infinity.
d) Mixed Flows
Steady, transonic flows contain both supersonic and subsonic regions. The
supersonic regions are hyperbolic in character while the subsonic regions are
elliptic.
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Studies in CFD – Summer 2012 - Homework 1
9.- Write down a brief explanation/description of the various types of numerical
grids used in CFD.
Structured (regular) grid -- Regular or structured grids consist of families of grid
lines with the property that members of a single family do not cross each other and
cross each member of the other families only once. This allows the lines of a given
set to be numbered consecutively. The position of any grid point (or control
volume) within the domain is uniquely identified by a set of two (in 2D) or three
(in 3D) indices, e.g. (i,j, k). This is the simplest grid structure, since it is logically
equivalent to
Block-structured grid -- In a block structured grid, there is a two (or more) level
subdivision of solution domain. On the coarse level, there are blocks which are
relatively large segments of the domain; their structure may be irregular and they
may or may not overlap. On the fine level (within each block) a structured grid is
defined. Special treatment is necessary at block interfaces.
Unstructured grids -- For very complex geometries, the most flexible type of grid
is one which can fit an arbitrary solution domain boundary. In principle, such grids
could be used with any discretization scheme, but they are best adapted to the finite
volume and finite element approaches. The elements or control volumes may have
any shape; nor is there a restriction on the number of neighbor elements or nodes.
In practice, grids made of triangles or quadrilaterals in 2D, and tetrahedra or
hexahedra in 3D are most often used.
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10.- Explain briefly the meaning of each of the following properties of numerical
solution methods; highlight the differences among them and explain why all of
them must be considered in CFD studies.
a) Consistency
For a method to be consistent, the truncation error must become zero when the
mesh spacing Δt  0 and/or Δxi  0. Truncation error is usually proportional to a
power of the grid spacing Δxi and/or the time step Δt. Some discretization methods
lead to truncation errors which are functions of the ratio of Δxi to Δt. In such a
case the consistency requirement is only conditionally fulfilled: Δxi and Δt must be
reduced in a way that allows the appropriate ratio to go to zero. Even if the
approximations are consistent, it does not necessarily mean that the solution of the
discretized equation system will become the exact solution of the differential
equation in the limit of small step size. For this to happen, the solution method has
to be stable.
b) Stability
A numerical solution method is said to be stable if it does not magnify the errors
that appear in the course of numerical solution process. For temporal problems,
stability guarantees that the method produces a bounded solution whenever the
solution of the exact equation is bounded. For iterative methods, a stable method is
one that does not diverge.
c) Convergence
A numerical method is said to be convergent if the solution of the discretized
equations tends to the exact solution of the differential equation as the grid spacing
tends to zero.
For linear initial value problems, the Lax equivalence theorem states that "given a
properly posed linear initial value problem and a finite difference approximation to
it that satisfies the consistency condition, stability is the necessary and sufficient
condition for convergence.
For non-linear problems the stability and convergence of a method are difficult to
demonstrate. Therefore convergence is usually checked using numerical
experiments. If the method is stable and if all approximations used in the
discretization process are consistent, we usually find that the solution does
converge to a grid-independent solution.
d) Conservation
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Since the equations to be solved are conservation laws, the numerical scheme
should also - on both a local and a global basis - respect these laws. This means
that, at steady state and in the absence of sources, the amount of a conserved
quantity leaving a closed volume is equal to the amount entering that volume. If the
strong conservation form of equations and a finite volume method are used, this is
guaranteed for each individual control volume and for the solution domain as a
whole. This is an important property of the solution method, since it imposes a
constraint on the solution error. If the conservation of mass, momentum and energy
are insured, the error can only improperly distribute these quantities over the
solution domain.
e) Boundedness
Numerical solutions should lie within proper bounds. Physically non-negative
quantities (like density, kinetic energy of turbulence) must always be positive;
other quantities, such as concentration, must lie between 0% and 100%. In the
absence of sources, some equations (e.g. the heat equation for the temperature
when no heat sources are present) require that the minimum and maximum values
of the variable be found on the boundaries of the domain.
f) Realizability
Models of phenomena which are too complex to treat directly (for example,
turbulence, combustion, or multiphase flow) should be designed to guarantee
physically realistic solutions.
g) Accuracy
Numerical solutions of fluid flow and heat transfer problems are only approximate
solutions. In addition to the errors that might be introduced in the course of the
development of the solution algorithm, in programming or setting up the boundary
conditions, numerical solutions always include three kinds of systematic errors:
 Modeling errors, which are defined as the difference between the actual flow
and the exact solution of the mathematical model;
 Discretization errors, defined as the difference between the exact solution of
the conservation equations and the exact solution of the algebraic system of
equations obtained by discretizing these equations, and
 Iteration errors, defined as the difference between the iterative and exact
solutions of the algebraic equations systems.
The ultimate goal is to obtain desired accuracy with least effort, or the maximum
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accuracy with the available resources.
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