External flows: Flow around bodies immersed in fluid

Examples:

• Air flow around (aerodynamics):
• buildings
• wind turbine blade
• airplanes
• cars
• Water flow around (hydrodynamics):
• offshore wind turbine foundation
• vessels
• submarines
• water turbine

Accurate description of external fluid forces: Key in optimizing designs with regards to, e.g.,

• Power generation (wind, hydro, wave)
• Fuel reductions (airplanes, cars, vessels)
• Structural strength (material usage)

Approaches

• Limited analytical techniques
• Experimental methods: Wind tunnels, towing tanks, wave bassins etc.
• Numerical methods (CFD): Large variation in fidelity

Categorization of bodies:

• 2-D bodies (infinitely long, constant cross-section)
• Axisymmetric bodies (cross-section rotated about symmetry axis)
• 3-D bodies

Note: Nominal 2-D bodies may need to be modelled as 3-D depending on the turbulence characteristics of the flow

Further categorization:

• Streamlined bodies (airfoils, racing cars) - little fluid disturbance
• Blunt/bluff bodies (stalled airfoils, buildings, parachutes) - significant fluid disturbance
• Generally less fluid resistance to streamlined bodies

External force on "stationary" body in flow with upstream velocity $U$

• Normal stresses due to pressure
• Wall shear stresses due to viscous effects

Pressure and wall shear stress distributions: Useful for detailed design and analysis

Usually we are mostly interested in the integrated stresses on the body, i.e., the resultant force

• Drag $D$: Component of the resultant force in the direction of the free-stream velocity $U$
• Lift $L$: Component of the resultant force normal to the free-stream velocity $U$

$p$ and $\tau_w$ vary in magnitude and direction: Contribute to both $D$ and $L$

$p$ and $\tau_w$ distributions are difficult to obtain (virtually only from CFD)

Often, we use dimensionless coefficients to estimate drag and lift forces:

• Drag coefficient: $$C_D = \dfrac{D}{\frac{1}{2} \rho U^2 A}$$
• Lift coefficient: $$C_L = \dfrac{L}{\frac{1}{2} \rho U^2 A}$$

Characteristic area $A$:

• Frontal area (projected in streamwise direction)
• Planform area (projected normal to streamwise direction)
• Other definitions (wetted surface, etc.)

Power consumption:

$$P = D \cdot U = \frac{1}{2} \rho U^3 C_D A$$ Scales with $U^3$!

Drag can be separated into two components:

• Drag caused by viscous shear stresses: friction drag
• Drag caused by pressure differences: pressure drag

Pressure (form) drag dominates for blunt bodies, while skin friction dominates for streamlined

Shape and Flow	Pressure (form) drag	Friction drag
	≈0%	≈100%
	≈10%	≈90%
	≈90%	≈10%
	≈100%	≈0%

Drag (physics)

Two bodies where characteristic lengths vary with an order of magnitude: Equal drag!

• Underlines the impact of streamlining
• Separation causes the drag!

Shape of body is important, but not the full story.

• Simple shapes $\neq$ simple flows
• Simplicity (or lack of) is governed by the flow characteristics
• Flow characteristics represented by dimensionless numbers:
• Reynolds number ($Re$): inertia / viscous
• Froude number ($Fr$): gravity / inertia
• Mach number ($Ma$): inertia / elasticity (compressibility)
• Weber number ($We$): inertia / surface tension
• Common orders of magnitude for charactheristic lengths and upstream velocities in external engineering flows yields:
• $10 < Re < 10^9$
• Rule of thumb: $Re > 100$ → flow dominated by inertial effects

Parallel flow to a flat plate at three different $Re = \{$$0.1$$,\,$$10$$,\,$$10^7$$\}$.

• Plate length: $l$
• Upstream velocity: $U$
• Boundary layer: $u < 0.99U$

Parallel flow to a flat plate at three different $Re = \{$$0.1$$,\,$$10$$,\,$$10^7$$\}$.

• Plate length: $l$
• Upstream velocity: $U$
• Boundary layer: $u < 0.99U$
• Large disturbance of flow

Parallel flow to a flat plate at three different $Re = \{$$0.1$$,\,$$10$$,\,$$10^7$$\}$.

• Plate length: $l$
• Upstream velocity: $U$
• Boundary layer: $u < 0.99U$

Parallel flow to a flat plate at three different $Re = \{$$0.1$$,\,$$10$$,\,$$10^7$$\}$.

• Plate length: $l$
• Upstream velocity: $U$
• Boundary layer: $u < 0.99U$
• Small viscid area
• Thin BL: $\delta_{99} \ll l$
• Wake region

Flow past a circular cylinder at (same) three $Re = \{$$0.1$$,\,$$10$$,\,$$10^7$$\}$.

• Diameter: $D$
• Upstream velocity: $U$
• Boundary layer: $u < 0.99U$

Flow past a circular cylinder at (same) three $Re = \{$$0.1$$,\,$$10$$,\,$$10^7$$\}$.

• Diameter: $D$
• Upstream velocity: $U$
• Boundary layer: $u < 0.99U$

Flow past a circular cylinder at (same) three $Re = \{$$0.1$$,\,$$10$$,\,$$10^7$$\}$.

• Diameter: $D$
• Upstream velocity: $U$
• Boundary layer: $u < 0.99U$
• Separation occurs
• Inertia too large to follow contour of body
• Recirculation region appears

Flow past a circular cylinder at (same) three $Re = \{$$0.1$$,\,$$10$$,\,$$10^7$$\}$.

• Diameter: $D$
• Upstream velocity: $U$
• Boundary layer: $u < 0.99U$
• Separation forms a turbulent wake
• Thin BL: $\delta_{99} \ll l$
• Recalling the costs of resolving turbulence - which part of this domain do you want to spent your bucks on resolving turbulence?
• For more info refer to (e.g.) papers on Detached Eddy Simulation (DES)

Closer inspection of the flow past a circular cylinder and sphere as a function of $Re$.

• Drag crisis (E): Sudden drop at $10^5 \lt Re \lt 10^6$

• Surface roughness can trigger turbulent BL at lower $Re$
• Typical golf balls: $10^4 \lt Re \lt 10^5$
• Why do golf balls have dimples?

Force coefficients: Drag, lift, added mass coefficients have been determined experimentally or numerically for many typical geometries and flow conditions

• Example: Drag coefficients for 2-D bodies
• Tables with force coefficients can be found in various standards, e.g.
  • DNV RP C205
• Or fluid mechanics textbooks:
• Hoerner (1965)
• Blevins (1984)

Hoerner (1965) & Blevins (1984)

Approaches to Turbulence Modeling

Hart et al. (2016)