8.4: Fluid Flow and Forces

1. Velocity and Force Measurements in fluids

   1. Physical properties

      1. Pressure -- force per unit area

         1. Result of collisions of fluid/gaseous particles with solid surfaces

         2. Pressure is always a relative term \[p_{system} = p_{absolute} = p_{atmospheric} + p_{gage} \] which can be drawn schematically (check lecture recording)

      2. Volume flow rate -- quantity of material moving through time

         1. Requires three length dimensions and time unit

         2. Material can subdivide through a control volume (or be additive if multiple sources)

         3. Has functional form \[ \dot{\underline{V}}   = \int \vec{V}\cdot \hat{n} dA \] where \(\vec{V}\) is the velocity vector perpendicular to (hence dot product of \(\hat{n}\)) to local incremental area \(dA\) (confirm with dimensional check)

         4. If velocity is uniform over an area, then algebraic form of \[ \dot{\underline{V}}_{in} = - |\vec{V}| A \] \[ \dot{\underline{V}}_{out} = + |\vec{V}| A \]

      3. Mass flow rate -- scale that incorporates fluid density \(\rho\) \[ \dot{m} = \int \rho\ \vec{V}\cdot \hat{n}\  dA \]

      4. Reaction forces in fluid flow

         1. Fluid must displace around solid object

         2. Aerodynamics uses high lift relative to drag to create flight

         3. Nature forms the most aerodynamic shape -- raindrop

         4. How does shape and velocity affect drag forces? Through dimensionless coefficient of drag: \[ c_{drag} = \frac{2 F_{drag}}{\rho V^{2} A} \]

   2. Conservation of Mass

      1. Tracking mass through Reynolds Transport Theorem

         1. Since mass of a system cannot be created or destroyed \(\frac{D m_{sys}}{Dt}=0\)

         2. Full equation \[ 0 = \frac{\partial }{\partial t} \int\limits_{CV} \rho d\underline{V} + \oint\limits_{CS} \rho\ \left(\vec{V}\cdot\hat{n}\right)\ dA \]

         3. Some noted features:

            • \(\int\limits_{CV} \rho\ d\underline{V}\) is total mass in the control volume

            • \(\int\limits_{CS} \rho\ \left(\vec{V}\cdot \hat{n}\right)\ dA\) is mass flow rate (\(\dot{m}\)) for area of interest

            • dot product automatically manages the signs for inflow (negative) and outflow (positive)

            • Total equation is only a scalar

            • Like nodal analysis from electrical engineering: larger diameter wire can carry more current -- but charge of electron can be different due to changing density

      2. Explicit assumptions/simplifications that must be stated (if used)

         1. Density is constant over the volume and/or area: \(\rho\int\limits_{CV} d\underline{V}\) and \(\rho\int\limits_{CS} \left(\vec{V}\cdot \hat{n}\right)\ dA\); constant can come out of the integral

         2. Steady mass flow conditions \(\Rightarrow \frac{\partial (\_\_)}{\partial t} = 0\); implies mass flow rate in equals mass flow rate out

         3. Uniform velocity over an area (comes out of integral): \(\rho \left(\vec{V}\cdot \hat{n}\right)\int\limits_{CS} dA = \rho \left(\vec{V}\cdot \hat{n}\right) A\)

         4. Velocity projected onto control surface area: when \(\vec{V}\ \|\ \hat{n}\) then \(\left(\vec{V}\cdot \hat{n}\right) = \pm V\) with sign associated with out and in, respectively

         5. Velocity components perpendicular to a control surface create no flux: \(\vec{V}\ \bot\ \hat{n} = 0\)

   3. Conservation of Linear Momentum

      1. Derivation:

         1. If \(B_{sys} = m\vec{V}\) (the momentum), then Reynolds Transport Theorem yields: \[ \frac{D m\vec{V}}{Dt} = \frac{\partial }{\partial t} \int\limits_{CV} \rho \vec{V} d\underline{V} + \oint\limits_{CS} \rho\ \vec{V}\left(\vec{V}_{\mathrm{rel}}\cdot \hat{n}\right)\ dA
            \]

         2. Recall Newton's Second Law of rigid body: \[ \sum \vec{F} = m\vec{a} = m\frac{d\vec{V}}{dt} \]

         3. Since the mass can change with time to affect momentum: \[ \sum \vec{F} = \frac{d (m\vec{V})}{dt} = m\frac{d\vec{V}}{dt} + \vec{V}\frac{dm}{dt}
            \]

         4. Thus forces can act on control volume and control surfaces: \[ \sum \vec{F} = \frac{\partial }{\partial t} \int\limits_{CV} \rho\ \vec{V}\ d\underline{V} + \oint\limits_{CS} \rho\ \vec{V}\ \left(\vec{V}_{\mathrm{rel}}\cdot\hat{n}\right)\ dA \]

      2. Notes about the equation:

         1. Made of three parts:

            Forces acting on drawn \(CV\) =

            Rate of accumulation of momentum in \(CV\)  +

            Flux of momentum in and out of \(CV\)

         2. Vector equation: all three directions embedded in single equation

            • \(\rho\  \vec{V}\  (\vec{V}\cdot\hat{n}) \Rightarrow \) Vector multiplied by 2 scalars

            • Forces in \(x\)-direction are function of \(u\)-component of velocity

         3. Momentum IN is negative while momentum OUT is positive

            • Sign of velocity vector is carried through though too

         4. Summation of forces are ALL acting on \(CV\)

            • Body forces (weight of stuff inside \(CV\))

            • Surface forces (pressure normal to surface, viscosity parallel to surface, etc.)

         5. All the assumptions from Conservation of Mass necessary to simplify to form \(\vec{F} = \dot{m}\vec{V}\)

      3. Exercise \(\PageIndex{1}\)

         The NIU wind tunnel has cross-sectional dimensions 18" by 18" and is calibrated by holding a 4" diameter sphere in the center of the test section. The incoming flow of air (\(\rho =2.3\times 10^{-3}\) slug/ft\(^{3}\)) is made uniform with velocity of 62 ft/s. At a point 8 inches downstream of the sphere, velocity measurements are found to be uniform at particular radial locations. The magnitude \(V_{center}\) is 4.7 ft/s and occurs only in an area of diameter 3 in immediately behind the sphere. An annulus (i.e., donut shape) area has a uniform velocity \(V_{fast}\) that goes from the \(D_{in}=3\) in to \(D_{out} = 10\) in. The remaining area of the wind tunnel's square cross-section still moves at 62 ft/s. The pressure and viscous forces are negligible and density is constant.

         

         Determine the uniform velocities that should be measured in the annulus region along with the total drag force necessary to hold the sphere in place.

         Answer

         \(V_{fast} = 67.67\) ft/s is the high speed flow in the annulus

         \(R_{x} = 58.658\) lb to hold the sphere in place

   4. Conservation of energy (Bernoulli equation)

      1. Tracking energy through combination of static, dynamic, and hydrostatic pressure

      2. Constant equation along a streamline \[p_{static} + \frac{1}{2} \rho V^{2} + \rho g h = p_{total} \]

      3. Dimensional change possible through algebraic manipulation \[\frac{P}{\rho g}+\frac{V^{2}}{2 g} + Z = h^{*} = \mathrm{Constant}\] when

         • fluid is incompressible (\(\rho\) is constant)

         • flow is steady: \(\frac{\partial}{\partial t}=0\)

         • flow is frictionless (\(\mu=0\))

         • along a streamline

         • pressure and gravity are the only forces

      4. Since fluid cannot turn a corner, but pressure can, height difference in a manometer fluid can relate the pressure difference and associated velocity \[ V_{flow} = \sqrt{\frac{2}{\rho_{air}} \rho_{manometer} g \Delta h} \]

2. Sensors

   1. Vane Anemometer

      1. Rotating blades serve as inverse to fan

      2. Convert rotational frequency to air speed

      3. Rated for minimum cut-in velocity

      4. Integrated temperature probe 

   2. Pitot Tube Anemometer

      1. Difference in stagnation versus static pressure leads to velocity

      2. Also observable in height difference of fluid (i.e., in of H\(_{2}\)O)

      3. Produce both analog and digital comparison

      4. May explore rotational dependence or velocity profile

   3. Force gage

      1. Mount different shapes to explore drag coefficient