[유체] Chap 2. Fundamentals of frictionless fluid flow

Chap 2. Fundamentals of frictionless fluid flow

2.1 Partial & Total differential

• Lagrangian description of flow : treats single particle’s motion
• Eulerian description of flow : perspective of Control Volume(CV)
• Conversion equation btw lagrangian and eulerian

\[{d \over dt} (\cdot) = {\partial \over \partial t} (\cdot) + (\vec{u} \cdot \nabla ) (\cdot)\]

• Total differential (Material / Substantial differential) : LHS
• $(\vec u \cdot \nabla) $ : Advection differential

2.2 Four fundamental flow lines

Streamline

• Tangents are parallel to velocity vector

\[{u \over \Delta x} = {v \over \Delta y} \rightarrow {dy \over dx} = {v \over u}\]

• No flux on normal direction
• Formula : ${dx \over u}= {dy \over v} = {dz \over w} = ds$, Integrate and eliminate s

Pathline

• Trajectory of a single particle

\[{dx \over dt} = u,\ \ {dy \over dt} = v\]

• Seperate by parameter, integrate and apply initial condition

Streakline

• Trajectory of particles that passed on certain point (Drops ink on certain position)

\[{dx \over dt} = u(t, \tau),\ \ {dy \over dt} = v(t, \tau)\]

• Also seperate by parameter, apply condition of streakline : initial position at time $t=\tau$
• Must specify the time : ie) streakline at time t=0

Timeline

• Line formed by marking a set of adjacent particle at same instant in time

2.3 Kinematic description

1. Translation $\vec u = (u,v,w)$
2. rotation : determined by angle change of diagonals of an element

\[\text{Rotation} = \theta = {\theta_1 +\theta_2 \over 2} = {1\over 2} \big({\partial v \over \partial x} - {\partial u \over \partial y}\big) = {1\over 2} (\nabla \times \vec u )_z = \omega_z\] \[\text{Magnitude of rotation } \omega = \sqrt{w_x^2+w_y^2+w_z^2}\] \[\text{Vorticity} = 2{\mathbf \omega} = \nabla \times \vec u\]

• Irrotational flow : when vorticity is zero
  1. Shear strain : rate of shrink / expand. Shrink is considered as positive

\[\text{Shear strain} = \theta_1 + \theta_2 = {\partial v \over \partial x} + {\partial u \over \partial y}\] \[\epsilon_{xy} = \epsilon_{yx} = {1\over 2} \big( {\partial v \over \partial x} + {\partial u \over \partial y} \big)\]

1. Linear strain

\[\nabla \cdot \vec u = {\partial u \over \partial x}+{\partial v \over \partial y}+{\partial w \over \partial z}\]

• If linear strain is zero, incompressible flow

2.4 System volume, Control volume

• System volume : Material volume, Tracking volume of a particle (Lagrangian perspective)
• Control volume : Fixed or moving volume. which is moving in assigned coordinate