© fox, pritchard, & mcdonald introduction to fluid mechanics chapter 5 introduction to...
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© Fox, Pritchard, & McDonald
Introduction to Fluid Mechanics
Chapter 5
Introduction to Differential Analysis of
Fluid Motion
© Fox, Pritchard, & McDonald
Main Topics
جرم بقای قانون تراکم بعدی دو جریان برای جریان تابع
ناپذیرسینماتیک مومنتوم معادله
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جرم بقای :قانونBasic Law for a System
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Conservation of MassDifferential CV and Taylor series
Infinitesimal control volumeof dimensions dx, dy, dz Area of right
face = dy dz
Mass flow rate throughthe right face of the
control volume
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Net mass flow rate into CV:
Net mass flow rate out of CV:
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After substitution,
Dividing through by volume dxdydz
Or, if we apply the definition of the divergence of a vector
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Conservation of Mass
Rectangular Coordinate System
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Conservation of Mass
Rectangular Coordinate System
“Del” Operator
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Conservation of Mass
Rectangular Coordinate System
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Conservation of MassCylindrical coordinates
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Conservation of MassSpecial Cases
Steady compressible flow
Cartesian
Cylindrical
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Conservation of Mass
Rectangular Coordinate System
Incompressible Fluid:
Steady Flow:
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EXE :A two-dimensional converging duct is being designed for a high-speed wind tunnel. The bottom wall of the duct is to be flat and horizontal, and the top wall is to be curved in such a way that the axial wind speed u increases approximately linearly from u1 " 100 m/s at section (1) to u2 " 300 m/s at section (2) .Meanwhile, the air density ρ is to decrease approximatelylinearly from ρ1 " 1.2 kg/m3 TO ρ2 " 0.85 .The converging duct is 2.0 m long and is 2.0 m high at section(1). (a) Predict the y-component of velocity, v(x, y), in the duct. (b) Plot theapproximate shape of the duct, ignoring friction on the walls. (c) How highshould the duct be at section (2), the exit of the duct?
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EXE 2:
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Conservation of Mass
Cylindrical Coordinate System
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Conservation of Mass
Cylindrical Coordinate System
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Conservation of Mass
Cylindrical Coordinate System
“Del” Operator
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Conservation of Mass
Cylindrical Coordinate System
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Conservation of Mass
Cylindrical Coordinate System
Incompressible Fluid:
Steady Flow:
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تراکم جریان برای جریان تابعTwo-Dimensional Flowبعدی 2ناپذیر
Stream Function
یاد آوری
خطوط جریان
ببریم را سرعت مولفه دومفهوم دل ψدر
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جریان خط طول در
d dx dyx y
جریان ψتابع خط طول دراست ثابت
Hence we can specify individual streamlines by their stream function values: ψ 5 0, 1, 2, etc. What is the significance ofthe ψ values?
دست به درد بهدبی حجمی آوردن
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the flow rate across AB is
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Stream Function for Two-DimensionalIncompressible Flow
Cylindrical Coordinates
Stream Function (r,)
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Motion of a Fluid Particle (Kinematics)Fluid Translation: Acceleration of a
Fluid Particle in a Velocity FieldFluid RotationFluid Deformation
• Angular Deformation• Linear Deformation
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برشی تنش تولید ها حرکت از یک کداممیکند؟
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( , , , )V V x y z t y
x
t0
t1
drr r particle path
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Fluid Translation: Acceleration of aFluid Particle in a Velocity Field مشتق
یا اصلیمشتق مادی
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جایی جابه شتاب محلی شتاب
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Motion of a Fluid Particle (Kinematics)
Fluid Translation: Acceleration of aFluid Particle in a Velocity Field
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Fluid Translation: Acceleration of aFluid Particle in a Velocity Field (Cylindrical)
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Fluid Rotation سیال چرخش
x1
u1(x3)u3(x1)
داریم؟ چرخش کی . باشیم داشته گشتاور بایدسطح بر عمود نیروهایشکل تغییر و شتاب باعثایجاد گشتاور و میشن طولیوتنشهای. نیروها نمیکنندبرای فقط که الزمه برشی
میده معنی ویسکوز جریان
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Rotation Definition of rotation
Time=t
x
y
ROTATION2z
d
dt
x
y
y x xV
y xV
x yV
x y yV
Assume Vy|x < Vy|x+x
and Vx|y > Vx|y+y
Time = t + t
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Fluid Rotation
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Ψ=0 if x=0 , 180 or r=R
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Motion of a Fluid Particle (Kinematics)
Fluid Deformation:• Angular Deformation
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Motion of a Fluid Particle (Kinematics)
Fluid Deformation:• Angular Deformation
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Motion of a Fluid Particle (Kinematics)
Fluid Deformation:• Linear Deformation
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Momentum Equation
Newton’s Second Law
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Momentum Equation
Forces Acting on a Fluid Particle
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Momentum Equation
Forces Acting on a Fluid Particle
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Momentum Equation
Differential Momentum Equation
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Momentum Equation
Newtonian Fluid: Navier-Stokes Equations
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Momentum Equation
Special Case: Euler’s Equation
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Computational Fluid Dynamics
Some Applications
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Computational Fluid Dynamics
Discretization