5 (fluid flow)
TRANSCRIPT
Fundamentals of fluid flow
Mohsen SoltanpourEmail: [email protected] URL: http://sahand.kntu.ac.ir/~soltanpour/
. . . : )(Velocity field ) (particles ) (solid . :
) (v x ) n = f n (t ) (v z ) n = hn (t
) (v y ) n = g n (t
n:*
. ) - **:(field approach
) v x = f ( x, y , z , t ) v y = g ( x, y , z , t ) v z = h ( x, y , z , t/http://sahand.kntu.ac.ir/~soltanpour
) (x,y,z ) (t *. ) (steady flow .** )(unsteady flow . :
) v x = f ( x, y , z ) v y = g ( x, y , z ) v z = h ( x, y , z
. ) (x,r, ) ( ) (: )v = v ( r , x = vr (r , x)er + v x (r , x)i
r 2 v = v x (r )i = vmax [1 ( ) ]i R
x
r ) (uniform ***./http://sahand.kntu.ac.ir/~soltanpour
.* :
x
**:z
0V
x
0V
xz
) (streamlines :/http://sahand.kntu.ac.ir/~soltanpour
) (pathlines . .
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dx = v xp dt dy = v yp dt dz = v zp dt
:
dA t )(streamtube . . . . ) dA (
v = vx i + v y j + vz k ds = dxi + dyj + dzk
V ds
0 = v ds
v ds :
dx dy dz = = vx v y vz) ( ) ( . ) (streakline ./http://sahand.kntu.ac.ir/~soltanpour
t < t1
t > t1
(Pathline) :
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t < t1
t > t1
(Streakline) :
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) ( . .
)0pathline (t=t
)0pathline (t>t
)0streakline (t=t/http://sahand.kntu.ac.ir/~soltanpour
)0streakline (t>t
: P ) ( . . A 0= t )0K (x0,y0,z t=t )1 P (x1,y1,z. A . t=t +t ) (B P B . C t=t +2t P C . 1 t A 1 A B 1 B C 1 C . P ) 1.(t=t)0 :M(x0,y0,z t=t+2t P )0 :L(x0,y0,z t=t+t P )0 :K(x0,y0,z t=t P )1 : P(x1,y1,z
) 1( t=t
) 2( t=t
) 3( t=t
1A 1C 2C 3C
1B 2B 3B
2A 3A
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C
B
A
2 t=t 3 t=t . B A C ) ( . . )1 P(x1,y1,z . 0= t )0 (x0,y0,z :
) x = f ( x0 , y0 , z0 , t ) y = g ( x0 , y0 , z0 , t
)(I
)1 P(x1,y1,z ) 0= t L K M P (:
) z = h( x0 , y0 , z0 , t
) x1 = f ( x0 , y0 , z0 , ) y1 = g ( x0 , y0 , z0 ,
)(II
) z1 = h( x0 , y0 , z0 , t P . 0 x0,y0,z ) (I ) (II t t ) t( P ) t( . t t P ) B A .(C t (t>t) t P t ) 1 t=t2 t=t 3.(t=t
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: )(Viewpoints 1 x1, y1, z ) v(x,y,z,t . .
) v x = f ( x, y , z , t ) v y = g ( x, y , z , t ) v z = h ( x, y , z , t . ) y(t) x(t ) z(t . )0( y(0) x )0( z 0= t . :
) v x = f (t ) v y = g (t
) v z = h(t/http://sahand.kntu.ac.ir/~soltanpour
t.
) .(a *. ./http://sahand.kntu.ac.ir/~soltanpour
: )(Acceleration of a flow particle . . :
zp
rp = x p (t )i + y p (t ) j + z p (t )k vp rp
= vp
drp
O
xy
= ap
dt dv p
= u p (t )i + v p (t ) j + w p (t )k
dt
=
d rp2
2 dt
= a xp (t )i + a yp (t ) j + a zp (t )k
) (rp . *./http://sahand.kntu.ac.ir/~soltanpour
) t y x z :
) v ( x, y , z , t
v v v v dx + dy + dz + dt = dv x y z t
.
dv Dv )(total derivative ) ( =a dt dt v dx v dy v dz v = + + + x dt y dt z dt t
dx , dy , dz v x , v y , v z . :
dt dt dt
v v v v v a = vx + vy + vz + + = v .v x y z t t )(acceleration of transport )(convective acceleration/http://sahand.kntu.ac.ir/~soltanpour
)(local acceleration
ax = vx
v x v v v + v y x + vz x + x x y z t v y v y v y v y a y = vx + vy + vz + x y z t v v v v az = vx z + v y z + vz z + z x y z t
:
. . zxQ 1v 2v
a
) (Q=cte x .
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1v2 > v v v v v v + vy + vz + 0 > = vx x a = vx y z t x x
) s (: v = v ( s, t ) = v( s, t )et dv v ds v s =a + . = dt s dt t v v v + =v s t
v = vet
) (normal and tangential coordinates :
v2 a = vet + en
dv dv ds 2 dv 1 dv = . =v = = at = v 2 ds dt ds dt ds 2v = an ) (osculari .
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( adjacent flowparticle) . dr = dxi + dyj + dzk t B A ( rotation rate) ( deformation rate) : B A B z dr dr dx dy dz y v= = i+ j+ k Ax
(Irrotational flow) :
v z dz z
v y
dt
v x dz z v z dx x v x dx xhttp://sahand.kntu.ac.ir/~soltanpour/
E z
dz
dt dt dt = vx i + v y j + vz k
B
dzA dxv y x dx
drdy
v z dy y
Dv x dy y
v y y
dy
C
A C
v vC = v A + dx x v x v y v z v = vC v A = dx + dxi + dxj dxk x x x x D E A ) (. ) (normal strain xx : AC
v x dx v x x = xx = dx x
= yy
v y y
. ) (dot ) ( . : = zz
v z z
) (normal strain rate 1/s. ii . AC z C : vy
x dx
dx
=
v y x
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v x dy v x y = dy y
AD :
(time rate of change of the shear angle) xy ) CAD t ( z :
v x ) ( = xy = yx + x yv v ) xz = zx = ( x + z z x
v y
CAE DAE :
v z ) ( = yz = zy + z y
v y
xx yx 2 zx 2
xy2 yy
zy2
2 yz 1 vi v j ( = ) + 2 2 x j xi zz
xz
) (strain rate tensor :
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) (rate of angular change of the sides . AC z C : vy
x dx
dx
=
v y x
AD z ) z( D :
v x dy v x y = dy y
CAD z ) AC (AD t : v
v 1 y [ ]) + ( x 2 x y
AC AD z )(z :
1 v y v x ) ( = z 2 x y
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x y :
1 v z v y ) ( = x 2 y z
1 v x v z ) ( = y 2 z x :
1 v z v y 1 v x v z 1 v y v x ( + )i ( +)j )k ( = 2 y z 2 z x 2 x y
i 1 = 2 x vx
) (vorticity vector curlv = rotv = v *: j k 1 1 ) = curlv = ( v 2 y z 2 v y vz
curl . 0 = ) (irrotational flow . 0 .**
)(rotational flow
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v y
0 = = curlv
x v z y v x z
v x 0= y v y 0= z v 0= z x
:
. . . ) (boundary layer .
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