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fluid mechanics
This is a mind map about fluid mechanics. Fluid mechanics is a discipline that studies the laws of mechanical movement of fluids and the laws of interaction between fluids and fluids, and between fluids and solids.
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fluid mechanics
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- Outline
fluid mechanics
introduction
definition
A discipline that studies the laws of fluid mechanical motion and the laws of interaction between fluids and fluids, fluids and solids
Research methods
On-site observation
laboratory simulation
theoretical analysis
Numeral Calculations
Continuous fluid medium model
The characteristic scale of flowing particles is much larger than the characteristic length of molecular motion.
The characteristic length of the problem discussed is much larger than the characteristic length of the fluid particle.
fluid particles
density
Knudsen number
As long as the Knudsen number is much less than 1, the continuum model of gases holds.
fluid properties
Stickiness (caused by momentum transport)
Experiment find-outs
Newton's law of viscosity
Meet: Newtonian fluid
Not satisfied: non-Newtonian fluids
liquid
The viscosity coefficient decreases as the temperature increases
gas
The viscosity coefficient increases as the temperature increases
A fluid that has no viscosity at all is called an ideal fluid
Thermal conductivity (due to energy transport)
Diffusion (due to mass transport)
Compressibility
Real fluids are compressible
compressible fluid
For liquids and low-speed flowing gases with small pressure differences, they can be regarded as incompressible fluids.
Fundamentals of Fluid Kinematics
Describe methods of fluid motion
Lagrangian method
For fluid particles
a, b, c, t are Lagrangian variables
Directly finding the partial derivative is the rate of change
Euler method
For fixed points in space
x,y,z,t are Euler variables
random derivative
steady flow
incompressible
homogeneous
The relationship between Lagrangian and Euler variables
Lagrangian becomes Euler
Solve a, b, c as single-valued functions of x, y, z, t
Euler to Lagrangian
geometric description method
trace
Lagrangian method
The trajectory of fluid particle motion
Each fluid particle has a trace, and different fluid particles have different traces.
The traces of all fluid particles form a family of curves
The intersection of the two surface equations after the Lagrangian variable expression cell t is the trace.
streamline
Euler method
The vector describing the velocity field is the streamline
At any time t, a curve imagined in the flow field. The tangent direction of any point on the curve coincides with the direction of the flow velocity at that point at that time.
Streamline differential equation
Cartesian coordinate system
cylindrical coordinate system
Spherical coordinate system
continuity equation
System and control body
system
Lagrangian method
System boundaries move with the system
There is no exchange of quality at the system boundary
There may be exchange of energy and interaction between the system and the outside world
control body
Euler method
On the control surface, there is an exchange of mass and energy
The control surface is stationary relative to the coordinate system
On the control surface, there is an interaction between controlling substances outside the body and controlling substances inside the body.
Reynolds transport formula
continuity equation
law of conservation of mass for fluids
There are no mass sources or sinks in the flow field.
Integral form of continuity equation
Differential form of continuity equation
Focus on flow field space points
Focus on fluid particles
Under different coordinate systems
Cartesian coordinate system
cylindrical coordinate system
Spherical coordinate system
Two special continuity equations
steady flow
incompressible
movement of fluid particles
Fluid micelle velocity decomposition theorem
Fluid micelle deformation
Relative linear expansion rate
relative volume expansion rate
Angular deformation
fluid micelle rotation
Vorticity
Irrotational motion and velocity potential
speed potential
At this time, there must be a scalar field function such that
simple shear motion
free vortex motion
forced vortex motion
Find velocity components from velocity potential
Cylindrical coordinates
Spherical coordinates
Velocity potential properties
The same velocity field can have different velocity potentials
Velocity potential is additive
In the irrotational flow field, the velocity loop along any curve is equal to the difference in velocity potential between the starting and end points.
Streamlines are orthogonal to equipotential surfaces
In a single connected area, the streamline cannot be closed, but in a multi-connected area, it can
Continuity equation of irrotational motion
The velocity potential is a harmonic function
Plane motion and stream functions of incompressible fluids
stream function
Introduction (in plane)
for incompressible fluids
There exists a scalar field function
Velocity field representation
Derivation
nature
The same velocity field has different flow functions, and the only difference between them is a constant related to t.
Stream functions are additive
There is no divergent motion in the plane. The flow rate through a certain curve is equal to the difference between the values of the stream functions at the two end points of the curve.
Isostream function lines are streamlines
The relationship between stream function and vorticity
If the fluid is irrotational, the flow function satisfies the Laplace equation and is a harmonic function.
Incompressible fluid motion, irrotational motion and complex potential
Introduction of complex potential
The complex potential is the characteristic function of plane divergence-free potential flow.
Plane divergent potential flow
Find the velocity potential and stream function
Find the speed
Find the velocity loop and flow rate
Several simple complex potentials
uniform flow
point source point sink
point vortex
dipole
flow around corners
Fundamentals of ideal fluid dynamics
force on fluid
quality force
Fluids whose mass force is only gravity are called gravitational fluids
surface force
normal stress
Shear stress
ideal fluid momentum equation
Integral form momentum equation for ideal fluid
Ideal fluid differential form momentum equation
Euler's equation
It can be divided into rectangular coordinate system, cylindrical coordinate system and spherical coordinate system.
Euler equation in rotating reference frame
ideal fluid energy equation
Ideal fluid integral form energy equation
Ideal fluid differential form energy equation
ideal fluid kinetic energy equation
internal energy equation
If the fluid pressure is distributed uniformly, the mechanical energy per unit mass of fluid remains unchanged.
Equation completeness and definite solution conditions
Equation completeness
There are four independent equations from the continuity equation and Euler's equation, but there are 5 unknown functions
incompressible homogeneous fluid
Positive pressure fluid
baroclinic fluid
Complete gas flow process adiabatic
Heat absorption in a complete gas flow process is caused only by heat conduction
Definite solution conditions
Initial conditions
Boundary conditions
Boundary conditions at infinity
Kinematic boundary conditions on the interface
Dynamic boundary conditions on the interface
One-dimensional motion of an incompressible ideal fluid
Integral form of Euler's equation
Bernoulli integral
Lambert equation
If the flow is steady
Lagrange integral
Ideal positive pressure fluid performs irrotational motion
In particular, for an incompressible ideal gravity fluid undergoing steady irrotational motion
Flow around a cylinder
Flow around a circular cylinder
Complex potential, velocity field and streamlines
Current sharing and dipole flow superposition
D'Alembert's fallacy
A cylinder moves in a stationary fluid without any resistance from the fluid.
Circular flow around a cylinder
Fundamentals of fluid vortex motion
Vortex motion concept
Description of vortex motion
Cartesian coordinate system
cylindrical coordinate system
Spherical coordinate system
vortex line
Vortex surface and vortex tube
Eddy flux
speed loop
The relationship between eddy flux and velocity circulation
Vortex kinematic properties
Helmholtz's first theorem
The vortex flux remains constant along the vortex tube
Velocity loop random derivative
Vortex Dynamic Properties
Vortex conservation
Kelvin's theorem (velocity potential conservation theorem)
Lagrange's theorem (theorem of the existence of velocity potential)
Helmholtz's second and third theorems
The creation, development and demise of vortices
Generation and diffusion of vortices in viscous fluids
Generation of vortices in baroclinic fluids
baroclinic fluid
Pieknes theorem
Vorticity field determines velocity field
Irrotational velocity field with scattered field
Dispersionless spin velocity field
Solve Poisson's equation
There are scattered and rotating velocity fields
Line vortex (vortex wire) induced velocity field (Bioshafar formula)
Rankin combination vortex
Velocity and pressure distribution outside the vortex core
Velocity and pressure distribution inside the vortex core