© Fox, Pritchard, & McDonald Introduction to Fluid Mechanics Chapter 5 Introduction to Differential...

© Fox, Pritchard, & McDonald

Introduction to Fluid Mechanics

Chapter 5

Introduction to Differential Analysis of

Fluid Motion


Main Topics

جرم بقای قانون تراکم بعدی دو جریان برای جریان تابع

ناپذیرسینماتیک مومنتوم معادله


جرم بقای :قانونBasic Law for a System


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


Net mass flow rate into CV:

Net mass flow rate out of CV:


After substitution,

Dividing through by volume dxdydz

Or, if we apply the definition of the divergence of a vector


Conservation of Mass

Rectangular Coordinate System




“Del” Operator


Conservation of MassCylindrical coordinates


Conservation of MassSpecial Cases

Steady compressible flow

Cartesian

Cylindrical




Incompressible Fluid:

Steady Flow:


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?


EXE 2:



Cylindrical Coordinate System




“Del” Operator




Incompressible Fluid:

Steady Flow:


تراکم جریان برای جریان تابعTwo-Dimensional Flowبعدی 2ناپذیر

Stream Function

یاد آوری

خطوط جریان

ببریم را سرعت مولفه دومفهوم دل ψدر


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


the flow rate across AB is


Stream Function for Two-DimensionalIncompressible Flow

Cylindrical Coordinates

Stream Function (r,)


Motion of a Fluid Particle (Kinematics)Fluid Translation: Acceleration of a

Fluid Particle in a Velocity FieldFluid RotationFluid Deformation

• Angular Deformation• Linear Deformation


برشی تنش تولید ها حرکت از یک کداممیکند؟


( , , , )V V x y z t y

x

t0

t1

drr r particle path


Fluid Translation: Acceleration of aFluid Particle in a Velocity Field مشتق

یا اصلیمشتق مادی


جایی جابه شتاب محلی شتاب


Motion of a Fluid Particle (Kinematics)

Fluid Translation: Acceleration of aFluid Particle in a Velocity Field


Fluid Translation: Acceleration of aFluid Particle in a Velocity Field (Cylindrical)


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


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


Fluid Rotation


Ψ=0 if x=0 , 180 or r=R



Fluid Deformation:• Angular Deformation



Fluid Deformation:• Linear Deformation


Momentum Equation

Newton’s Second Law


Momentum Equation

Forces Acting on a Fluid Particle


Momentum Equation

Differential Momentum Equation


Momentum Equation

Newtonian Fluid: Navier-Stokes Equations


Momentum Equation

Special Case: Euler’s Equation


Computational Fluid Dynamics

Some Applications


Computational Fluid Dynamics

Discretization

© Fox, Pritchard, & McDonald Introduction to Fluid Mechanics Chapter 5 Introduction to Differential...

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