Momentum Heat Mass Transfer

1
Momentum Heat Mass Transfer
MHMT6
Bernoulli equation. Boundary layer.
Solutions of the Navier-Stokes equations in
limiting cases. Engineering Bernoulli equation.
Boundary layer theory. Flow along a plate. Karman
integral theorem.
Rudolf Žitný, Ústav procesní a zpracovatelské
techniky CVUT FS 2010
2
Navier Stokes - high Re
MHMT6
Let us consider the 2D Navier Stokes equation
describing for example a steady state fluid flow
around a body (diameter D) at incoming velocity
U. The following equation represents a momentum
balance in the incoming flow direction and should
have been used for prediction of drag forces
Introducing dimensionless variables Xx/D, Yy/D,
Uxux/U, Uyuy/U, ReUD/? gives dimensionless
equation
It seems to be obvious that with the increasing
velocity (with increasing Re) the Navier Stokes
equation reduces to the Eulers equation and the
last term of viscous forces is less and less
important (all dimensionless variables X,Y,Ux,Uy
are supposed to be of the unity order, with the
exception of Re).
3
dAlemberts paradox
MHMT6
DAlemberts paradox. Analytical solutions based
upon Euler equation indicate, that the resulting
force (integrated along the whole surface of
body) should be zero. It was a great challenge
for the best brains of 19th century (Lord
Rayleigh, Lord Kelvin, von Karman) to explain
the controversy between experience (scientists
knew about the quadratic increase of drag forces
with velocity U) and theory represented either by
the Stokes solution for drag on a sphere
(cD24/Re) or steady state solutions of Eulers
equations predicting zero drag for high Reynolds
number. Suspicion was focused to possible
discontinuities/ instabilities of potential
flows, and to wake, resulting to explanation of
many important phenomena, for example the Karman
vortex street (see previous lecture, or read von
Karman paper, T.Karman Über den Mechanismus des
Wiederstandes, den ein bewegter Körper in einer
Flussigkeit erfahrt. Nachrichte der K.
Gesellschaft der Wissenschaften zu Göttingen
Mathematisch-physikalische Klasse. (1911)
509-517).
Quite different view was suggested by Ludwig
Prandtl during conference in 1904
Prandtl L. Über Flussigkeitsbewegung bei sehr
kleiner Reibung. Verhandlungen d.III
Internat.Math.Kongress, Heidelberg 8.-13.August
1904, B.G.Teubner, Leipzig 1905, S.485-491
These 7 pages caused similar revolution like the
Einsteins relativity theory. Prandl realised
that the dimensional analysis can be misleading
and that the last term on the right hand side of
the previous equation cannot be neglected even
for infinitely large Re, because viscous fluid
sticks at wall and very large velocity gradients
exist in a thin boundary layer. The whole flow
field is to be separated to an inviscid region
and to a boundary layer, described by parabolised
NS equations.
4
Boundary layer
MHMT6
Outside the boundary layer the velocity field is
described by the Euler equation. For inviscid
incompressible flow the relationship between
velocities and pressure are expressed by
Bernoulli theorem
5
Boundary layer
MHMT6
Inspection analysis must be carried out with a
great care taking into account relative magnitude
of individual terms in the Navier Stokes
equations (and to distinguish magnitudes in
longitudinal (1) and transversal (?) direction).
.terms of the order ? can be neglected
6
Boundary layer
MHMT6
Equations describing boundary layer are reduced
to
Remark While the original Navier Stokes
equations are elliptic, the boundary layer
equations are parabolic, which means that we can
describe an evolution of boundary layer in the
direction x (this coordinate plays a similar role
as the time coordinate in the time evolution
problems). This is a great advantage because
marching technique enables step by step solution
and it is not necessary to solve the whole
problem at once.
The last equation shows that the pressure is
constant in the transversal direction and its
value is determined by Bernoulli equation applied
to outer (inviscid) region,
therefore the momentum equation in the
x-direction is
7
Boundary layer - separation
MHMT6

• These equations are the basis of Prandtls
  boundary layer theory, which offers the following
  two important results
• Viscous (friction) drag forces, called skin drag
  can be predicted. As soon as there is no
  separation of boundary layer from the surface the
  outer flow is not affected by the presence of
  boundary layer. The outer flow can be therefore
  solved in advance separately giving (via
  Bernoullis equation) pressure and boundary
  conditions for the boundary layer region.
• Form drag. Point of separation of the boundary
  layer on highly curved surfaces (cylinders,
  spheres, airfoils at high attack angles) can be
  predicted too. The separation occurs at the point
  with adverse pressure gradient, and the separated
  boundary layer forms a Helmholtz discontinuity
  surface. Behind the discontinuity is formed a
  wake (dead fluid region) increasing the form
  drag (drag caused by pressure imbalances)
  significantly.

8
Boundary layer - Plate
MHMT6
Probably the most important question is the
thickness of boundary layer ?(x). Only
qualitative answer follows from the boundary
layer equation (parallel flow along a plate with
zero pressure gradient)
approximated very roughly as
giving
This preliminary conclusion is qualitatively
correct boundary layer thickness increases with
square root of distance, kinematic viscosity and
decreases with increasing free stream velocity.
See also the theory of PENETRATION DEPTH.
9
Boundary layer - Plate
MHMT6
Little bit more precise analysis is based upon
LINEAR velocity profile across the boundary
layer. Integral balances of a rectangular control
volume (height H)
Continuity equation
Integral balance in the x-direction
10
Boundary layer - Plate
MHMT6
Linear velocity profile results in not very
precise solution
Cubic velocity profile is much better giving
Exact formulation of differential equation of
boundary layer was presented by Blasius
(Prandtls student), however this ordinary
differential equation requires numerical
solution. Anyway, knowing approximations of
velocity profiles, it is possible to calculate
viscous stresses upon the plate and therefore the
drag force. For linear velocity profile
more acccurate Blasius solution
linear profile
11
Von Karman Integral theorem
MHMT6
Prandtl equations of boundary layer are partial
differential equations. Karman theorem derives
from these equations ordinary differential
equation, suitable for approximate solutions.
Hopper
12
Integral equations
MHMT6
Continuity equation integrated across the
boundary layer
Momentum equation
Elimination of transversal component using
continuity equation
13
Integral equations
MHMT6
Integrated momentum equation can be rearranged to
Displacement thickness ?
Momentum thickness ?
Displacement of surface corresponding to the same
flowrate of ideal fluid
In a similar way the energy integral equation can
be derived (multiply momentum equation by ux and
integrate)
Energy thickness ?
Dissipation integral
14
Integral equations
MHMT6
Integral momentum equation holds for laminar as
well as turbulent flow
The only one differential equation is not enough
for solution of 3 variables displacement
thickness, momentum thickness and shear stress.
Approximate solutions are based upon assumption
of similarity of velocity profiles in the
boundary layer.
15
Integral equations
MHMT6
The simplest example is laminar boundary layer at
parallel flow along a plate. In this case
Uconstant and integral momentum balance reduces
to
Better approximation than the previously analyzed
linear velocity profile uxUy/? is a cubic
velocity profile, because the cubic polynomial
with 4 coefficients can respect 3 necessary
boundary conditions corresponding to laminar
flow zero velocity at surface y0, prescribed
velocity U and zero stress dux/dy at y?.
16
Integral equations
MHMT6
Substituting this velocity profile to definition
of ? and ?w?dux/dy
Please mention the fact that the momentum
thickness ? is much less than the boundary
layer thickness ?
Karman integral balance reduces to the ordinary
differential equation for thickness of boundary
layer
Solution is the previously presented (but not
derived) result
17
Integral equations
MHMT6
Turbulent boundary layer is described by the same
integral equation, but it is not possible to use
the same velocity profile (this is not true that
duz/dy is zero at y?) and first of all the
turbulent wall shear stress cannot be expressed
in the same way like in laminar layer.
Brutal simplification based upon linear velocity
profile, and simplified Prandtls model of
turbulence (???2U2) gives linearly increasing
boundary layer thickness
In reality ? increases more slowly, with the
exponent 0.8 ( ) this
prediction is based upon Blasius formula for
friction factor (see textbook Sestak et al
Prenos hybnosti a tepla (1988), p.94). More
accurate result (see Schlichting, Gersten
Boundary layer theory, Springer, 8th edition
2000) is
G only weakly depends on Re and limiting value is
1
18
Integral equations
MHMT6
Typical values of boundary layer thickness ? and
thickness of laminar sublayer for plate (remark
even in the turbulent flow there is always a thin
laminar layer adjacent to wall, see next lecture)
Fluid Um/s L m Re ? mm ?lam mm
air 10 1
air 50 1 3.3?106 8 0.4
water 1 2 2?106 17 1
water 2 5 107 39 0.6
19
EXAM
MHMT6
Boundary layers
20
What is important (at least for exam)
MHMT6
Bernoullis equation (outer inviscid region)
Boundary layer equation in the direction of flow
21
What is important (at least for exam)
MHMT6
Thickness of boundary layer (plate)
Drag of parallel flow on plate