Fluid Dynamics and the
Applications of Differential
Equations
By Shi-Jeisun Xie, Summer 2011
Fluid Mechanics
• Fluid mechanics is a discipline under continuum mechanics that
studies fluids (liquids, gases, plasma) and the associated forces
• Fluid mechanics can be divided into:
o Fluid statics or hydrostatics: the study of fluids at rest and
effects of forces on fluid equilibrium
o Fluid dynamics: the study of fluids in motion and effects of
forces on fluid flow
• Follows the continuum assumption:
o Fluids are considered to be continuous entities instead of
discrete particles. Properties such as density, pressure,
temperature, and velocity are well-defined at infinitesimally
small points and vary continuously throughout.
Fluid Dynamics
• Fluid dynamics is the study of fluid flow and the effects of forces
on fluids in motion
• This sub-discipline can be separated into hydrodynamics (liquids
in motion) and aerodynamics (gases in motion)
• Based on the conservation laws of mass, linear momentum, and
energy
• Expressed using the Reynolds transport theorem, which essentially
states:
o The final quantity inside a control volume is equal to the initial
quantity plus the amount that enters the control volume minus
the amount that leaves the control volume
Characteristics and Models of Fluid
Flow - Compressibility
• Compressible fluids experience changes in density due to changes in
pressure or temperature; all fluids are compressible to some degree
o Compressible fluids use more general compressible flow
equations
• In the case that density changes are insignificant and negligible, the
flow can be modeled as incompressible
o Incompressible fluids do not experience changes in density as they
move in the flow fluid. That is,
o All single-component liquids at constant temperature and gases at
fluid velocity less than the speed of sound with insignificant
temperature gradients behave as incompressible fluid
o For most biological systems, fluids are often treated as
incompressible due to relatively constant pressure and temperature
o For acoustic problems, fluids are often treated as compressible to
examine effects of sound compression waves
Characteristics and Models of Fluid
Flow - Viscosity
• Viscous fluids are significantly affected by fluid friction in respect to
fluid flow and have high resistance to shear stress
• Inviscid fluids, or ideal fluids, experience no resistance to shear stress
and have no viscosity
o Inviscid flow often use the Euler equations and Bernoulli's equation
• In regards to kinematics and dynamics, we are often concerned with the
ratio of inertial forces to viscous forces, as represented by the Reynolds
number Re
o If Re is very low (>>1), inertial forces are negligible compared to
viscous forces, and the fluid is in Stokes flow (creeping flow)
o If Re is very high, viscous forces are negligible compared to inertial
forces, and the flow can be modeled as inviscid
• Near solid boundaries, viscosity generally cannot be ignored
o These regions utilize the boundary layer equations for computation
Viscous Flow vs. Inviscid Flow
More viscous
Less viscous
Inviscid (Ideal)
Viscous
Characteristics and Models of Fluid
Flow - Laminar and Turbulent Flow
• Fluid flow is laminar when there is no disruption among parallel
fluid layers in the flow field
o Laminar flow has high momentum diffusion and low
momentum convection. That is, when the fluid flows past a
solid surface, momentum diffuses across the boundary layer
o Flows with low Re (below 2100) are usually laminar
• Fluid flow is turbulent when there are non-deterministic and
"random" changes in property
o Turbulent flow is by definition unsteady
o Flows with high Re (above 4000) are usually turbulent
• Under steady flow, pressure, shear stress, velocity, and other fluid
properties at a given point do not change with time. Otherwise, the
flow is unsteady
Laminar Flow vs. Turbulent Flow
Turbulent
Laminar
Characteristics and Models of Fluid
Flow - Stress and Strain Relationship
• Newtonian fluids show a linear relationship between stress and
rate of strain
o For Newtonian fluids, viscosity is the coefficient of
proportionality and is constant for a particular fluid
o In fluid dynamics, viscosity is commonly used to characterize
shear properties
o Most fluids of mid/low viscosity are Newtonian. e.g. water, air
• Non-Newtonian fluids show a nonlinear relationship between
stress and rate of strain
o For non-Newtonian fluids, stress and strain rate are dependent
on numerous factors, and a constant viscosity cannot be defined
o Many highly viscous fluids and polymer solutions are nonNewtonian
Newtonian Flow vs. Non-Newtonian
Flow
• Green: Newtonian
• Red & Blue: Non-Newtonian
Example of a Newtonian
fluid - Water
Example of a nonNewtonian fluid - Paint
The Continuity Equations
• Continuity equations are differential equations that describe
relations of some conserved quantity (mass, energy, momentum,
etc.)
• They are based on the physical laws of conservation
• The general form is
where φ is the a quantity, t is time, ∇ is the divergence operator or
gradient operator, f is flux (flow rate), s is generation or removal
rate of the quantity. The equations of fluid dynamics can be
expanded and expressed in rectangular (Cartesian), cylindrical,
and spherical coordinates, but the Cartesian representation is the
most commonly used. In Cartesian coordinates, ∇ is defined with
unit vectors u as
• For conserved quantities, s=0, and so
The Substantial Derivative
• Describes rate of change with respect to time of a quantity while
in motion with velocity v
• In respect to fluid dynamics, it describes the rate of change with
respect to time of a quantity moving along a path in accordance
with fluid flow
• Also known as the material derivative, Stokes derivative, etc.
• The operator for the substantial derivative is defined as:
where x is a scalar or vector
Conservation of Mass
• The mass continuity equation (mass is conserved, so s=0) is:
where ρ is mass density and v is velocity (ρv is "flux")
• Because of the expansion
the above equation can be written using the substantial derivative:
Conservation of Mass - Incompressible
Fluids
• For incompressible fluids, ρ = constant, thus
becomes
in which case
Derivation of the Mass Continuity
Equation
• [Rate of accumulation of mass in control volume] = [Flow rate of
mass into control volume] - [Flow rate of mass from control
volume]
o For a cubic control volume, the mass = (density)(volume
element ΔxΔyΔz)
o Mass flow rate =(density)(cross-sectional area)(local velocity)
• Assuming that (for each direction) mass enters the control volume
at x (for a surface of constant x and area ΔyΔz) and leaves at x+Δx,
we can write the above equation as
Derivation of Mass Continuity Equation
(cont.)
Dividing both sides by ΔxΔyΔz gives
Using the definition of the derivative
and taking the limit as Δx, Δy, Δz approach 0 gives
Derivation of Mass Continuity Equation
(cont.)
Using the gradient operator for Cartesian coordinates
and replacing the unit vectors with ρv, we obtain the divergence of the mass flow rate per unit area
Thus, in accordance to the general continuity equation,
or
Conservation of Linear Momentum
• [Rate of momentum accumulation] = [rate of momentum flow in] [Rate of momentum flow out] + Σ Forces
• The modeling for linear momentum is similar to that for mass, but
whereas mass was a scalar, momentum is a vector
o For a cubic control volume, the momentum = (momentum per
unit volume)(volume element ΔxΔyΔz)
• We can write the above equation as
or
Derivation of Linear Momentum
Continuity Equation
Taking the limit as Δx, Δy, Δz approach 0 gives
where dV is the differential volume element dxdydz
Again, using the gradient operator for Cartesian coordinates ∇, we obtain
and
Derivation of Linear Momentum
Continuity Equation (cont.)
Expanding the ∇expression and derivative of momentum into , respectively
gives
where the third and fourth terms in the brackets both represent the product of
velocity and the conservation of mass and thus are equal to zero, so
Conservation of Linear Momentum Body Forces
• Forces acting on the control volume are either body forces or
surface forces, thus
• Body forces are those that act on the entire body (as oppose to
contact forces), such as gravity or electromagnetic forces. For
fluid dynamics, we only consider gravity, thus
and by taking the limit as Δx, Δy, Δz approach 0, we obtain
Conservation of Linear Momentum Surface Forces
• Surface forces are those that act on the surfaces of the control
volume, such as pressure and viscous stress
o Pressure act normal to the surface and is the stress on the fluid
at rest
o Viscous stresses arise from the motion of the fluid and act both
normally and tangentially to the surface
• The total stress tensor is the sum of the pressure and the viscous
stresses, or
where σ is the stress tensor, τ is the viscous stresses, p is pressure and
I is the identity matrix. The negatives sign arises because a
compressive stress is considered to be negative and pressure is a
positive quantity
Conservation of Linear Momentum Stress Tensors
• Pressure is isotropic: it is the same in all directions for a given
point
o Pressure is normal to every surface and directed inward
• Viscous stresses are deviatoric: they are not generally the same in
all directions for a given point
o Viscous stress has nine components, with three directional
stresses on each constant surface
o e.g. for a surface of constant x, the fluid stresses acting in the x,
y, z directions are τxx τxy τxz, respectively
o e.g. the sum of surface forces in the x direction are τxx, τyz, τzx, p
• Expressing the total stress tensor in rectangular coordinates thus
gives
Conservation of Linear Momentum Derivation of Surface Forces
• The surface force arises from a gradient in the stress tensor (∇·σ),
thus it is derived from the difference in pressure and the sum of the
viscous forces for each respective direction
o By convention, the fluid on the face with the greater algebraic
value of the defining space variable (e.g. x+Δx) exerts positive
stress on the face that has the lesser value (e.g. x). This results
in a (conventional) negative p factor. Thus, for the x direction
Likewise
Conservation of Linear Momentum Derivation of Surface Forces (cont.)
Following Fsx as the example
we can write the right side expression as a product of the volume element and force per
unit volume, or
Taking the limit as Δx, Δy, Δz approach 0, we obtain
Applying this form form to Fsy and Fsz and rearranging the terms, we obtain
Conservation of Linear Momentum Derivation of Surface Forces (cont.)
Because τx=τxx+τyx+τzx (and similarly for τy, τz), using the definition of the gradient
operator in Cartesian coordinates (see previous slides), we obtain
since
we obtain
and
Conservation of Linear Momentum General Form
Incorporating the force term into
we obtain
dividing both sides by the differential volume element and rearranging the left side
expression yields
• This is the general equation for the conservation of linear
momentum and is another form of Newton's second law of
motion, expressed per unit of volume
• In some forms, ρg is replaced by Fb, a general term for body force
• In the absence of fluid motion, v=0, τ=0, thus we obtain a form of
the equation of fluid statics
The Navier-Stokes Equations
• The Navier-Stokes equations are differential equations that
describe the motion of fluid
o They state that the changes in momentum depends only
on external pressure and internal viscous stresses
• The general form of the Navier-Stokes equation is
which is just another expression of the conservation of linear
momentum using the substantial derivative for the left side
expression and accounting ρg with Fb
• Application of the Navier-Stokes equation requires
information on the stress tensor term τ, which depends on
the specific type of fluid flow
Forms of the Navier-Stokes Equations
• Different fluid types require analysis using different forms of the
Navier-Stokes equation
• The stress tensor generally requires information on the viscosity of
fluid flow, and as such often deal with Newtonian fluids and fluids
with predictable relationships between stress and strain rate
• Newtonian fluids follow general relationship
where μ is the viscosity constant
• Understanding that viscous stress is symmetric for most fluids
The Navier-Stokes Equations Incompressible Newtonian Fluids
• Using Newton's law of viscosity and the symmetric identity of the
viscous stress tensor, we are able to develop the relationship
• Noting that
vector form
obtain
is the same as the transpose of
and that in
can be written as the velocity gradient ∇v, we
where (∇v)T is the transpose of ∇v.
• Because the fluid is incompressible, it has been shown earlier
that ∇·v=0, and because the stress tensor is symmetric, ∇v=(∇v)T,
so the viscous stress term becomes
and the Navier-Stokes equation for incompressible Newtonian fluids
becomes
Incompressible Newtonian Fluids Derivation of the Viscous Stress Term
Using the constitutive relationship
we obtain for the viscous stress of the x momentum direction
Thus, the viscous stress term of the Navier-Stokes equation becomes
Given that for an incompressible fluid,
we thus obtain
with similar results and derivation for the y and z momentum directions
The Navier-Stokes Equations Compressible Newtonian Fluids
• For compressible Newtonian fluids, the Navier-Stokes equation is
similar to that of incompressible fluids, with some exceptions
o The viscous stress tensor includes an additional term for the
bulk viscosity for the compressible particles of the fluid, which
does exist for incompressible flow due to the nonexistence of
flow divergence
• The bulk viscosity applies only when the viscous stress acts
normally to the surface (i.e. when i = j for τij)
o The term for the bulk viscosity is
where μv is the bulk for second coefficient of viscosity, δij is the
Kronecker delta ( = 1 when i = j, = 0 when i ≠ j)
Compressible Newtonian Fluids Derivation of the Bulk Viscosity Factor
• Because δij = 1 only for τxx, τyy, τzz, and δij = 0 for all other τij
• By incorporating the bulk viscosity factor with the viscous stress
tensor found for incompressible fluids, we obtain
and the Navier-Stokes equation for compressible fluids becomes
Computational Fluid Dynamics - An
Application of Navier-Stokes
• Computational fluid dynamics (CFD) use algorithms, numerical
methods, and computer calculations to analyze fluid flow problems
• In CFD follows a general basic procedure:
o Preprocessing: the physical boundaries and boundary conditions
are defined, the control volume is divided into discrete cells, and
the equations for physical modeling are defined
o Processing: the simulation runs and iteratively solves the defined
equations as in or not in steady-state
o Postprocessing: the solutions are further analyzed and visualized
• CFD can be used to model fluid flow, especially turbulent flow, by
finding or approximating solutions to the Navier-Stokes equations
o Because turbulent flow is associated with a wide range of length
and time scales, resolution of these scales can be computationally
costly depending on the finesse and accuracy of the model
Examples of CFD
• Some examples of CFD for turbulence modeling include
o Direct numerical simulation (DNS) - solves the Navier-Stokes
equations and resolves the entire range of length and time scales
for turbulence
 Allows for simulation of turbulent flow, but is extremely
expensive and memory-intensive at higher Reynolds numbers
(computational cost is proportional to Re3)
o Reynolds-averaged Navier-Stokes (RANS) modeling - models
fluid flow and Reynolds stresses using time-averaged equations
incorporating Reynolds decomposition to approximate solutions
to the Navier-Stokes equations
 Reynolds decomposition separates a quantity into its average
and fluctuating components
Visual Examples of CFD and DNS
DNS analysis of the turbulent heat flux
DNS analysis of turbulent mean velocity
DNS analysis of turbulent kinetic energy
Sources
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Cambridge Univ. Pr., 2000.
• Fox, Rodney O. Computational Models for Turbulent Reacting
Flows. Cambridge, U.K.: Cambridge UP, 2003.
• Munson, Bruce Roy, Donald F. Young, and T. H. Okiishi.
Fundamentals of Fluid Mechanics. Hoboken, NJ: J. Wiley & Sons,
2006.
• Segel, Lee A., and G. H. Handelman. Mathematics Applied to
Continuum Mechanics. Philadelphia: Society for Industrial and
Applied Mathematics, 2007.