Flow Around Immersed Objects Incompressible Flow

Goals Describe forces that act on a particle in a fluid. Define and quantify the drag coefficient for spherical and non- spherical objects in a flow field. Define Stokes’ and Newton’s Laws for flow around spheres.

Flow Around Objects There are many processes that involve flow through a porous medium such as a suspension of particles:

Flow Around Objects There are many processes that involve flow around objects - some are more interesting than others:

Forces Dynamic Fk results from the relative motion of the object and the fluid (shear stress) Static Fs results from external pressure gradient (Fp) and gravity (Fg).

Dynamic Forces For flow around a submerged object a drag coefficient Cd is defined: U0 is the approach velocity (far from the object), ρ is the density of the fluid, A is the projected area of the particle, and Cd is the drag coefficient analogous to the friction factor in pipe flow (keep this in mind).

Projected Area Spherical Particle The projected area used in the Fk is the area “seen” by the fluid. Spherical Particle

Projected Area Cylinder For objects having shapes other than spherical, it is necessary to specify the size, geometry and orientation relative to the direction of flow. Cylinder Axis perpendicular to flow Rectangle Axis parallel to flow Circle

Drag Coefficient The drag coefficient, like the friction factor in pipes depends on the Reynolds number D is particle diameter or a characteristic length and ρ and μ are fluid properties.

Drag Coefficient Recall: Stokes’ Law for Creeping Flow Around Sphere For slow flow around a sphere and Re<10 Recall: Stokes’ Law for Creeping Flow Around Sphere

Drag Coefficient

Why Different Regions? As the flow rate increases wake drag becomes an important factor. The streamline pattern becomes mixed at the rear of the particle thus causing a greater pressure at the front of the particle and thus an extra force term due to pressure difference. At very high Reynolds numbers completely separate in the wake. Streamline separation

Drag Coefficient Adjustment Russian Shkval Torpedo

Determining Flow Fields

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Static Forces Static forces exist in the absence of fluid motion. They include the downward force of gravity and the upward force of buoyancy that results from the gravity induced pressure gradient in the z-direction

Total Force What is the direction of Fk? The gravity and buoyancy forces on an object immersed in liquids do not generally balance each other and the object will be in motion. What is the direction of Fk? It is always opposite to the direction of particle motion

Equilibrium When a particle whose density is greater than that of the fluid begins to fall in response to the force imbalance, it begins to accelerate (F=ma). As the velocity increases the viscous drag force also increases until all forces are in balance. At this point the particle reaches terminal velocity.

Terminal Velocity General Expression: If the particle has a uniform density, the particle mass is Vprp and Use: Falling ball viscometer to measure viscosity

Settling Velocity Stokes’ Region: The settling (terminal) velocity of small particles is often low enough that the Reynolds number is less than unity (Cd = 24/Re). Newton’s Region: Between 1000<Re<200,000 Cd = 0.44 Note: Intermediate flow requires iteration

Criterion for Settling Regime Reynolds number is a poor criteria for determining the proper regime for settling. We can derive a value K that depends solely on the physical parameters K < 2.6 Stokes’ Law K > 68.9 Newton’s Law

Example A cylindrical bridge pier 1 meter in diameter is submerged to a depth of 10m in a river at 20°C. Water is flowing past at a velocity of 1.2 m/s. Calculate the force in Newtons on the pier.

Example Fig. 7.3 gives Cd ≈ 0.35 Projected Area = DL = 10 m2

Example Estimate the terminal velocity of limestone particles (Dp = 0.15 mm, r = 2800 kg/m3) in water @ 20°C.

Example Guess Re = 4 Cd = 16.0 – from Figure 7.6

Example Guess Re = 2 Cd = 22 Ut = 0.013 m/s Re = 1.9

10 Minute Problem Tiger Woods is practicing putting golf balls on a cruise ship, he makes a slight miscalculation and the ball rolls off the “green” and falls into the ocean. Assuming the ball quickly attains its terminal velocity and the descent is defined by the Newton’s law flow regime, how long does it take the ball to hit the ocean floor 300 m below ?   Golf ball data: Diameter = 43 mm Weight = 45 grams Density = 1.16 g /cm3 Seawater data: Density = 1.025 g /cm3 Viscosity = 0.01 g / cm sec