mmme2047 lecture notes dimensional analysis 2023-02-06

uni/mmme/2xxx/2047_thermodynamics_and_fluid_dynamics/dimensional_analysis.md

@@ -3,9 +3,12 @@ author: Akbar Rahman
 date: \today
 title: MMME2047 // Dimensional Analysis
 tags: [ mmme2047, uni, fluid_dynamics, dimensional_analysis ]
-uuid: 
+uuid: 483f818f-3f67-4cde-bbd6-d9f93c89ca47
 ---
 
+[Lecture slides](./lecture_slides/T4 - Dimensional analysis - with solutions.pdf)  
+[Lecture notes](./lecture_notes/dimensional analysis 2018-2019.pdf)
+
 In lab tests it is not always possible to use the actual scale of the prototype,
 actual flow speed, or actual fluid.
 In these cases a model is used.
@@ -25,12 +28,19 @@ How to make sure a prototype and a scale model are physically similar
   linear scale ratioi.
 
   This includes surface roughness (e.g. a 10x smaller model have 10x smaller roughness)
-- Kinematic similarity ---velocities at corresponding points in the two flows are in the same direction and related by a constant scale factor in magnitude
+- Kinematic similarity ---velocities at corresponding points in the two flows are in the same direction and related by a constant scale factor in magnitude. Flow regimes (laminar, turbulent, compressible, etc.) must be the same.
 - Dynamic similarity --- requires that the magnitude ratio of any two forces in one system must be the same as the magnitude ratio of the corresponding forces in the other system
 
 Kinematic and Dynamic similarity are ensued by equality of the governing
 nondimensional parameters.
 
+As a general rule, dynamic and kinematic similarity are ensured if:
+
+- for compressible flow, prototype and model Reynolds and Mach number, and specific heat ratio, are correspondingly equal
+- For incompressible flow with no free surface, prototype and model Reynolds numbers are equal
+- For incompressible flow with a free surface, Reynolds and Froude numbers are equal (may also require equality
+  of Weber and cabiation number as well)
+
 # Dimensions and Units
 
 There are four basic dimensions for fluid dynamics:
@@ -142,7 +152,7 @@ Important in flows with interfaces (e.g. gas-liquid).
 
 $$\text{We} = \frac{rho U^2 L}{\sigma}$$
 
-where $\sigma$ is the surface tension coeffecient.
+where $\sigma$ is the surface tension coefficient.
 
 Represents ratio of inertial to capillary forces.
 Important to flows with strong surface tension effects (e.g. droplets,
@@ -155,9 +165,74 @@ $$\text{St} = \frac{fL}{U}$$
 where $f$ is frequency.
 
 Important in flows with velocity oscillations.
+$\text{St} \approx 0.21$ for $200 < \text{Re} < 10^5$.
 
 ## Mach Number
 
 $$\text{Ma} = \frac U a$$
 
+When $\text{Ma} > 0.3$, the flow should be considered compressible.
 
+# Nondimensional Momentum Equation (From Navier-Stokes)
+
+$$\frac{\del \pmb{V*}}{\del t*} + \pmb{V*} \cdot (\Del* \pmb{V*} = -\Del* p* + \frac{1}{\text{Re}}\Del*^2\pmb{V*}$$
+
+Therefore for large values of Reynolds number, the viscous forces become negligible.
+
+# Kinematic Similarity
+
+An airfoil model and prototype are geometrically similar, with $1:\alpha$ length ratio:
+
+- $x_m = \frac{x_p}{\alpha}$, $y_m = \frac{y_p}{\alpha}$
+- $u_m$ at $(x_m, y_m)$ must have the same direction at $u_p$ at $(x_p, y_p)$
+- $\frac{u_p}{u_m} = \Beta$ is constant at homologous points
+
+![](./images/vimscrot-2023-02-06T09:46:38,665212789+00:00.png)
+
+# Dynamic Similarity
+
+Dynamic similarity is similarity of forces.
+Two systems have dynamic similarity when:
+
+- they are geometrically similar
+- identical kind of forces are parallel and related by a constant scale factor at corresponding homologous point
+
+#### Example
+
+Consider two homologous points around two geometrically similar airfoils:
+
+Assume there are 3 forces acting: inertia, friction, and pressure.
+These forces must form a closed polygon: $\pmb{F_i} = \pmb{F_f} + \pmb{F_p}$.
+
+![](./images/vimscrot-2023-02-06T10:08:03,569393092+00:00.png)
+
+$F_{f,p}$ and $F_{f,m}$ must be parallel and the same applies for the other two forces.
+The force magnitude ratios must be related by a constant scale factor, therefore
+this means the triangles must be geometrically similar.
+
+We can estimate $F_i$ and $F_f$ (lecture slides p54-55):
+
+$$F_i ~ \rho L^2U^2$$
+
+$$F_f ~ \mu U L$$
+
+Therefore:
+
+$$\frac{F_i}{F_f} ~ \frac{\rho U L}{\mu} = \text{Re}$$
+
+This means that at homologous points, the magnitude ratio of inertia/viscosity  in one system is the
+same as that in the other system if the Reynolds number is the same.
+
+# Consequences of Incomplete Similarity
+
+In example 7 (lecture slides p57), it is not possible to achieve complete dynamic similarity.
+
+- In the case of the example, tests are usually run with water, due to its convenience
+- This violates Reynolds number similarity, but this does not matter much as Froude number is
+  dominant parameter in free surface flows.
+- Reynolds number of the model will be smaller and the model's data will be extrapolated to the desired
+  Reynolds number:
+
+  ![](./images/vimscrot-2023-02-06T10:42:58,992360702+00:00.png)
+
+- Extrapolation increases uncertainty and it is left to the engineer to judge the validity of the data