ME 231

1
ME 231 Thermofluid Mechanics I Navier-Stokes
Equations














2
Computational Fluid Dynamics (AE/ME 339)
K. M. Isaac MAEEM Dept., UMR

• Equations of Fluid Dynamics, Physical Meaning of
  the terms.
• Equations are based on the following physical
  principles
• Mass is conserved
• Newtons Second Law
• The First Law of thermodynamics De dq - dw,
  for a system.

3
Computational Fluid Dynamics (AE/ME 339)
K. M. Isaac MAEEM Dept., UMR
Control Volume
Analysis The governing equations can be obtained
in the integral form by choosing a control volume
(CV) in the flow field and applying the
principles of the conservation of mass, momentum
and energy to the CV.












4
Computational Fluid Dynamics (AE/ME 339)
K. M. Isaac MAEEM Dept., UMR












5
Computational Fluid Dynamics (AE/ME 339)
K. M. Isaac MAEEM Dept., UMR

• Consider a differential volume element dV in the
  flow field. dV is small enough to be considered
  infinitesimal but large enough to contain a large
  number of molecules for continuum approach to be
  valid.
• dV may be
• fixed in space with fluid flowing in and out of
  its surface or,
• moving so as to contain the same fluid particles
  all the time. In this case the boundaries may
  distort and the volume may change.

6
Computational Fluid Dynamics (AE/ME 339)
K. M. Isaac MAEEM Dept., UMR
Substantial derivative (time rate of change
following a moving fluid element)













7
Computational Fluid Dynamics (AE/ME 339)
K. M. Isaac MAEEM Dept., UMR

The velocity vector can be written in terms of
its Cartesian components as where u u(t,
x, y, z) v v(t, x, y, z) w w(t, x, y, z)












8
Computational Fluid Dynamics (AE/ME 339)
K. M. Isaac MAEEM Dept., UMR




_at_ time t1 _at_ time t2 Using Taylor series


 






9
Computational Fluid Dynamics (AE/ME 339)
K. M. Isaac MAEEM Dept., UMR



The time derivative can be written as shown on
the RHS in the following equation. This way of
writing helps explain the meaning of total
derivative.    









10
Computational Fluid Dynamics (AE/ME 339)
K. M. Isaac MAEEM Dept., UMR



We can also write









11
Computational Fluid Dynamics (AE/ME 339)
K. M. Isaac MAEEM Dept., UMR





where the operator can now be seen to
be defined in the following manner.







12
Computational Fluid Dynamics (AE/ME 339)
K. M. Isaac MAEEM Dept., UMR

The operator in vector calculus is defined
as     which can be used to write the total
derivative as  











13
Computational Fluid Dynamics (AE/ME 339)
K. M. Isaac MAEEM Dept., UMR



Example derivative of temperature, T









14
Computational Fluid Dynamics (AE/ME 339)
K. M. Isaac MAEEM Dept., UMR

A simpler way of writing the total derivative is
as follows











15
Computational Fluid Dynamics (AE/ME 339)
K. M. Isaac MAEEM Dept., UMR

The above equation shows that and
have the same meaning, and the latter form is
used simply to emphasize the physical meaning
that it consists of the local derivative and the
convective derivatives.   Divergence of Velocity
(What does it mean?)   Consider a control volume
moving with the fluid. Its mass is fixed with
respect to time. Its volume and surface change
with time as it moves from one location to
another.











16
Computational Fluid Dynamics (AE/ME 339)
K. M. Isaac MAEEM Dept., UMR







Insert Figure 2.4





17
Computational Fluid Dynamics (AE/ME 339)
K. M. Isaac MAEEM Dept., UMR

The volume swept by the elemental area dS during
time interval Dt can be written as     Note
that, depending on the orientation of the surface
element, Dv could be positive or negative.
Dividing by Dt and letting 0 gives the
following expression.











18
Computational Fluid Dynamics (AE/ME 339)
K. M. Isaac MAEEM Dept., UMR



The LHS term is written as a total time
derivative because the fluid element is moving
with the flow and it would undergo both the local
acceleration and the convective
acceleration.   The divergence theorem from
vector calculus can now be used to transform the
surface integral into a volume integral.









19
Computational Fluid Dynamics (AE/ME 339)
K. M. Isaac MAEEM Dept., UMR
If we now shrink the moving control volume to an
infinitesimal volume, , , the above
equation becomes     When the
volume integral can be replaced by on the RHS
to get the following.   The divergence of
is the rate of change of volume per unit
volume.












20
Computational Fluid Dynamics (AE/ME 339)
K. M. Isaac MAEEM Dept., UMR











21
Computational Fluid Dynamics (AE/ME 339)
K. M. Isaac MAEEM Dept., UMR
Continuity
Equation    Consider the CV fixed in space.
Unlike the earlier case the shape and size of the
CV are the same at all times. The conservation of
mass can be stated as   Net rate of outflow of
mass from CV through surface S time rate of
decrease of mass inside the CV












22
Computational Fluid Dynamics (AE/ME 339)
K. M. Isaac MAEEM Dept., UMR



The net outflow of mass from the CV can be
written as     Note that by convention
is always pointing outward. Therefore can be
() or (-) depending on the directions of the
velocity and the surface element.









23
Computational Fluid Dynamics (AE/ME 339)
K. M. Isaac MAEEM Dept., UMR



Total mass inside CV   Time rate of increase of
mass inside CV (correct this equation)     Conse
rvation of mass can now be used to write the
following equation    See text for other ways
of obtaining the same equation.









24
Computational Fluid Dynamics (AE/ME 339)
K. M. Isaac MAEEM Dept., UMR
Integral form of the conservation of mass
equation thus becomes












25
Computational Fluid Dynamics (AE/ME 339)
K. M. Isaac MAEEM Dept., UMR












26
Computational Fluid Dynamics (AE/ME 339)
K. M. Isaac MAEEM Dept., UMR
An infinitesimally small element fixed in space












27
Computational Fluid Dynamics (AE/ME 339)
K. M. Isaac MAEEM Dept., UMR
Net outflow in x-direction





Net outflow in y-direction


Net outflow in z-direction

      Net mass flow




28
Computational Fluid Dynamics (AE/ME 339)
K. M. Isaac MAEEM Dept., UMR
volume of the element dx dy dz mass of
the element r(dx dy dz)





Time rate of mass increase

Net rate of outflow from CV time rate of
decrease of mass within CV




or


29
Computational Fluid Dynamics (AE/ME 339)
K. M. Isaac MAEEM Dept., UMR
Which becomes






The above is the continuity equation valid for
unsteady flow Note that for steady flow and
unsteady incompressible flow the first term is
zero.


Figure 2.6 (next slide) shows conservation and
non-conservation forms of the continuity
equation. Note an error in Figure 2.6 Dp/Dt
should be replace with Dr/Dt.




30
Computational Fluid Dynamics (AE/ME 339)
K. M. Isaac MAEEM Dept., UMR












31
Computational Fluid Dynamics (AE/ME 339)
K. M. Isaac MAEEM Dept., UMR
Momentum equation












32
Computational Fluid Dynamics (AE/ME 339)
K. M. Isaac MAEEM Dept., UMR
Consider the moving fluid element model shown in
Figure 2.2b Basis is Newtons 2nd Law which says
F m a Note that this is a vector equation. It
can be written in terms of the three cartesian
scalar components, the first of which becomes
Fx m ax Since we are considering a fluid
element moving with the fluid, its mass, m, is
fixed. The momentum equation will be obtained by
writing expressions for the externally applied
force, Fx, on the fluid element and the
acceleration, ax, of the fluid element. The
externally applied forces can be divided into two
types












33
Computational Fluid Dynamics (AE/ME 339)
K. M. Isaac MAEEM Dept., UMR

• Body forces Distributed throughout the control
  volume. Therefore,
• this is proportional to the volume. Examples
  gravitational forces,
• magnetic forces, electrostatic forces.
• Surface forces Distributed at the control volume
  surface. Proportional
• to the surface area. Examples forces due to
  surface and normal stresses.
• These can be calculated from stress-strain rate
  relations.
• Body force on the fluid element fx r dx dy dz
• where fx is the body force per unit mass in the
  x-direction
• The shear and normal stresses arise from the
  deformation of the fluid
• element as it flows along. The shape as well as
  the volume of the fluid
• element could change and the associated normal
  and tangential stresses
• give rise to the surface stresses.

34
Computational Fluid Dynamics (AE/ME 339)
K. M. Isaac MAEEM Dept., UMR












35
Computational Fluid Dynamics (AE/ME 339)
K. M. Isaac MAEEM Dept., UMR
The relation between stress and rate of strain in
a fluid is known from the type of fluid we are
dealing with. Most of our discussion will relate
to Newtonian fluids for which Stress is
proportional to the rate of strain For
non-Newtonian fluids more complex relationships
should be used. Notation stress tij indicates
stress acting on a plane perpendicular to the
i-direction (x-axis) and the stress acts in the
direction, j, (y-axis). The stresses on the
various faces of the fluid element can written
as shown in Figure 2.8. Note the use of Taylor
series to write the stress components.












36
Computational Fluid Dynamics (AE/ME 339)
K. M. Isaac MAEEM Dept., UMR
The normal stresses also has the pressure term.




Net surface force acting in x direction








37
Computational Fluid Dynamics (AE/ME 339)
K. M. Isaac MAEEM Dept., UMR






   






38
Computational Fluid Dynamics (AE/ME 339)
K. M. Isaac MAEEM Dept., UMR






   






39
Computational Fluid Dynamics (AE/ME 339)
K. M. Isaac MAEEM Dept., UMR






           






40
Computational Fluid Dynamics (AE/ME 339)
K. M. Isaac MAEEM Dept., UMR
The term in the brackets is zero (continuity
equation) The above equation simplifies to






Substitute Eq. (2.55) into Eq. (2.50a) shows how
the following equations can be obtained.
   






41
Computational Fluid Dynamics (AE/ME 339)
K. M. Isaac MAEEM Dept., UMR
The above are the Navier-Stokes equations in
conservation form.





           







42
Computational Fluid Dynamics (AE/ME 339)
K. M. Isaac MAEEM Dept., UMR
For Newtonian fluids the stresses can be
expressed as follows





           







43
Computational Fluid Dynamics (AE/ME 339)
K. M. Isaac MAEEM Dept., UMR

• In the above m is the coefficient of dynamic
  viscosity and l is the second
• viscosity coefficient.
• Stokes hypothesis given below can be used to
  relate the above two
• coefficients
• l -
  2/3 m
• The above can be used to get the Navier-Stokes
  equations in the following
• familiar form

44
Computational Fluid Dynamics (AE/ME 339)
K. M. Isaac MAEEM Dept., UMR







 
 





45
Computational Fluid Dynamics (AE/ME 339)
K. M. Isaac MAEEM Dept., UMR







 
The energy equation can also be derived in a
similar manner.





46
Computational Fluid Dynamics (AE/ME 339)
K. M. Isaac MAEEM Dept., UMR
The above equations can be simplified for
inviscid flows by dropping the terms involving
viscosity.












47

• 
  Summary
• Apply conservation equations to a control volume
  (CV)
• As the CV shrinks to infinitesimal volume, the
  resulting partial
• differential equations are the Navier-Stokes
  equations
• Taylor series can be used to write the variables
• The total derivative consists of local time
  derivative and convective
• derivative terms
• In incompressible flow, divergence of velocity
  is a statement of
• the conservation of volume
• Need surface and body forces to write the
  momentum equation
• Surface forces are pressure forces, forces due
  to normal stresses
• and forces due to shear stresses
• Body forces are due to weight, magnetism and
  electrostatics
• Momentum equation is a vector equation. Can be
  written in terms
• of its components.

48

• 
  Summary
• Stresses are indicated by a plane and a
  direction, respectively, by two subscripts
• For Newtonian fluids, stresses are proportional
  to the rates of strain
• Stokes hypothesis is used to relate the first
  and second coefficients of viscosity
• The resulting equations are the Navier-Stokes
  equations
• In order to solve the equations, they must be
  simplified for the problem
• you are considering (e. g., boundary layer, jet,
  airfoil flow, nozzle flow)