[유체] Chap 4. Dimensional Analysis & Similarity

Chap 4. Dimensional Analysis & Similarity

4.1 Basic dimensions

• Mass
• Length
• Time
• Temperature
• Luminous intensity (Candela)
• Current
• Mole

Dimensional homogenity

• The dimension of each term in one equation is homogeneous
• e.g) bernoulli equation:

\[{P\over \rho} + gh + {1\over 2} v^2 = \text{const}\]

every dimension is $[L^2T^{-2}]$

4.2 Similarity

1. Geometric Similarity (Length, shape)
2. Kinematic Similarity (Velocity, Acceleration)
3. Dynamic Similarity (Force, Moment)

4.3 Buckinghem’s Pi theorem

• Number of dependent variables is $m$
• Number of independent variables is $n$
• Number of nondimensional variables is $m-n$

4.4 Key-nondimensional Numbers

Physical interpretation

• Inertia : $F=ma, \ \ \rho L^3\cdot L/T^2 = \rho L^2U^2$
• Viscous friction force : $F = \mu {\partial u \over \partial y} \times \text{Area} = \mu U L$
• Gravity force : $F = mg = \rho L^3 g$
• Pressure force $F = P\times \text{Area} = PL^2$

Nondimensional numbers

• Reynolds number (ratio of inertia over viscous frictional force)
• Froude number (squared root of inertia over gravitational force)
• Euler number (Ratio of pressure force over inertia)

\[\text{Reynolds number } Re \sim {\text{Inertia} \over \text{Viscous frictional force}} = {\rho L^2U^2 \over \mu UL} = {\rho U L \over \mu} = {UL \over \nu}\] \[\text{Froude number } Fr \sim \sqrt{\text{Inertia} \over \text{Gravitational force}} = \sqrt{\rho U^2L^2 \over \rho L^3 g} = \sqrt{U^2 \over gL} = {U \over \sqrt {gL}}\] \[\text{Euler number } Eu \sim {\text{pressure force} \over \text{Inertia}} = {PL^2 \over \rho L^2U^2} = {\Delta p \over \rho U^2}\]

More nondimensional number

• Cavitation number $Ca = {p-p_v \over {1\over 2} \rho U^2}$
• Weber number : $We = {\rho U^2L \over \sigma}$
• Strauhal number $St = {fL\over U}$
• Keulegan-Carpenter number $Ke = {UL \over T}$
• Frantl number $Fr= {\nu \over \alpha}$

4.5 Drag force on arbitary body

• Drag force is caused by : pressure force (form drag) + viscous frictional force

Drag coefficient $C_D$

\[C_D = {\text{Drag} \over {1\over 2} \rho U^2 S}\]

• $S$ : Projected area
• Direction of drag is same as the direction of flow

Lift coefficient $C_L$

\[C_L = {\text{Lift} \over {1\over2}\rho U^2 S_L}\]

• $S_L = \text{coord} \times \text{span}$ (coord : without considering AoA
• Lift force is acting on body perpendicular to the direction of flow (without considering AoA)

Measuring total drag force on a real ship

• Total drag can be decomposed with :
  • Pressure drag(form drag) : function of Reynolds no.
  • Friction drag : function of Reynolds no. dominent in $C_T$
  • Wave Resistance : function of Froude no.

2D method

\[C_T = C_F(Re) + C_R(Fr)\]

• To measure frictional drag coefficient, consider the coefficient is equivalent with the “drag coefficient of equivalent flat plate” (to minimize the form drag)

3D method

\[\begin{aligned}C_T &= C_F(Re) + C_W(Fr) + Cp(Re) \\ &=C_F(Re) + \kappa C_F(Re) + C_W(Fr) \\ &=(1+\kappa) C_F(Re) + C_W(Fr)\end{aligned}\]