equivalent Darcy friction factor for the multiphase flow uu i i = for all a r
r i i i i i i n x x = = 1 Fluid Mechanics 3-101 1999 by CRC
Press LLC (3.8.14) where (3.8.15) This homogeneous model permits
direct evaluation of all three components of axial p
(their thickness is greatly exaggerated in the figure). Separation may
occur in the region of increasing pressure on the rear of the body
(points S); after separation boundary layer fluid no longer remains in
contact with the surface. Fluid that was in th
negative for expansion. Let us define a pressure coefficient Cp, as where
q1 is the dynamic pressure and is equal to Equation (3.7.36) then gives
(3.7.37) Equation (3.7.37) states that the pressure coefficient is
proportional to the local flow deflection.
(1982). Equation (3.7.26) can also be written in terms of pressure as
follows: (3.7.27) Nozzle Flow Using the area relations, we can now plot
the distributions of Mach number and pressure along a nozzle. Figure
3.7.9 shows pressure and Mach number distrib
section (open side facing downstream) 1.20 a Data from Hoerner, 1965.
b Based on ring area. > ~ Fluid Mechanics 3-77 1999 by CRC Press LLC
Friction and Pressure Drag: Bluff Bodies Both friction and pressure
forces contribute to the drag of bluff bodies (s
+ - ( ) ( ) + - + + ( ) - ln ln g g g g g g
ss M M M 21 1 1 1 0 1 0 1 0 1 -= = < < > > for for for cT cT T T p p 01 02
01 02 = = or 3-86 Section 3 1999 by CRC Press LLC Let us now apply
the entropy change relation across the shock using the stagnation
co
462(2 0 - 0 1H)(9 81) H H 1 2 3 2 1 2 3 2 . sec . . . . , m solve for m 3 eff
0.94 Normal depth: m solve m 3 Q y y y y n n n n = = ( ) + 2 5
1 0 0 025 2 0 2 0 2 0 2 0 1 3 2 . sec . . . . . sin . , 1.14 Critical depth: A y
solve m bQ g y c c c = - = ( )
a normal shock wave at the test section Mach number. In practice, the
diffuser gives lower than expected recovery as a result of viscous losses
caused by the interaction of shock wave and the boundary layer which
are neglected here. The operation of super
simultaneously: gas/vapor, solids, single-liquid phase, multiple
(immiscible) liquid phases. Every possible combination has been
encountered in some industrial process, the most common being the
simultaneous flow of vapor/gas and liquid (as encountered in
component of lift acting in the flow direction increases drag; the
increase in drag due to lift is called induced drag. The effective aspect
ratio includes the effect of planform shape. When written in terms of
effective aspect ratio, the drag of a finite
below. Laminar Boundary Layers. A reasonable approximation to the
laminar boundary layer velocity profile is to express u as a polynomial in
y. The resulting solutions for d and tw have the same dependence on x
as the exact Blasius solution. Numerical res
LLC Basic relationships between these and related parameters are
(3.8.1) (3.8.2) (3.8.3) (3.8.4) (3.8.5) In most engineering calculations, the
above parameters are defined as average quantities across the entire
flow area, Ac. It should be noted, however,
transfer. References Abbott, I.H. and von Doenhoff, A.E. 1959. Theory of
Wing Sections, Including a Summary of Airfoil Data. Dover, New York.
Fox, R.W. and McDonald, A.T. 1992. Introduction to Fluid Mechanics, 4th
ed. John Wiley & Sons, New York. Hazen, D
1 2 2 1 1 = + + M M M M 3-88 Section 3 1999 by
CRC Press LLC Quasi-One-Dimensional Flow In quasi-one-dimensional
flow, in addition to flow conditions, the area of duct also changes with
x. The governing equations for quasi-one-dimensional flow can be
writ
equations can be written as (3.7.8) Equations (3.7.8) are applicable to a
general type of flow; however, for a calorically perfect gas, we can use
the relations p = rRT and h = cpT to derive a number of equations
relating flow conditions downstream of the
3.6.4, transition occurs at Rex = 500,000; the dashed line represents the
drag coefficient at larger Reynolds numbers. A number of empirical
correlations may be used to model the variation in CD shown in Figure
3.6.4 (Schlichting, 1979). Extending the lam
derived coefficients, discussed below. Drag coefficient is defined as
FIGURE 3.6.3 Boundary layer flow with presssure gradient (thickness
exaggerated for clarity). Fluid Mechanics 3-75 1999 by CRC Press LLC
(3.6.9) where 1/2rV2 is dynamic pressure and A i
Cd bg H where Cd = = dimensionless weir coefficient 1 2 3 2 , Sharpcrested: for Broad-crested: for C H Y L Y C H L d d + < < < 0 564 0
0846 0 07 0 462 0 08 0 33 . . . . . . Vee-notch, angle 2q q : Q 0 44 g H
for 10 < q 50 1 2 5 2 . tan dy dx S S V b gA V
occurs almost immediately after the minimum pressure at about 25%
chord aft the leading edge. Transition can be delayed by shaping the
profile to maintain a favorable pressure gradient over more of its
length. The U.S. National Advisory Committee for Aero
sections depend on Reynolds number and angle of attack between the
chord line and the undisturbed flow direction. The chord line is the
straight line joining the leading and trailing edges of the airfoil (Abbott
and von Doenhoff, 1959). As the angle of at
and computer power continues to be made, but the role of the
experimentalist likely will remain important for the foreseeable future.
Computational Fluid Dynamics (CFD) Computation of fluid flow requires
accurate mathematical modeling of flow physics and
obtain a first approximation by applying potential flow theory to
calculate the flow field around the object. Much effort has been
devoted to calculation of velocity distributions over objects of known
shape (the direct problem) and to determination of sh
at some velocity V and corresponding Mach number M. If this gas is
brought isentropically to stagnation or zero velocity, the pressure and
temperature which the gas achieves are defined as stagnation pressure
p0 and stagnation temperature T0 (also called
boundary layers and delaying separation and stall. Suction and Blowing
Suction removes low-energy fluid from the boundary layer, reducing the
tendency for early separation. Blowing via high-speed jets directed
along the surface reenergizes low-speed bound
Triangle Park, NC. 3-70 Section 3 1999 by CRC Press LLC 3.6 External
Incompressible Flows Alan T. McDonald Introduction and Scope
Potential flow theory (Section 3.2) treats an incompressible ideal fluid
with zero viscosity. There are no shear stresses; pr
assumption no longer remains valid as M1 . Let us now examine why
the flow ahead of a normal shock wave must be supersonic even
though Equations (3.7.8) hold for M1 < 1 as well as M1 > 1. From the
second law of thermodynamics, the entropy change across th
= 1 d dp t r r T T p = 1 t r r s s p = 1 a p a p s s
2 = = r r or p rg = constant 3-82 Section 3
1999 by CRC Press LLC where g is the ratio of specific heats at constant
pressure and constant volume, R is the gas constant, and T is the
temperature. For a
within the boundary layer. For zero pressure gradient, shear forces
alone can never bring the particle to rest. (Recall that for laminar and
turbulent boundary layers the shear stress varied as 1/x1/2 and 1/x1/5,
respectively; shear stress never becomes z
wall shear stress varies as 1/x1/5. Approximate results for laminar and
turbulent boundary layers are compared in Table 3.6.2. At a Reynolds
number of 1 million, wall shear stress for the turbulent boundary layer
is nearly six times as large as for the la