Determinación de las fuerzas de impacto en una placa plana con el método del control del volumen
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Overview
Fuente: Ricardo Mejía-Alvarez y Hussam Hikmat Jabbar, Departamento de ingeniería mecánica, Universidad Estatal de Michigan, East Lansing, MI
El propósito de este experimento es demostrar las fuerzas en cuerpos como el resultado de cambios en el ímpetu linear del flujo alrededor de ellos usando una formulación del volumen de control [1, 2]. El análisis del volumen de control se centra en el efecto macroscópico de flujo en sistemas de ingeniería, en lugar de la descripción detallada que puede lograrse con un análisis diferencial. Cada una de estas dos técnicas tienen un lugar en la caja de herramientas de un analista de ingeniería, y deben considerarse complementarios en lugar de enfoques concurrentes. En términos generales, análisis del volumen control dará al ingeniero una idea de las cargas dominantes en un sistema. Esto le dará a él/ella una sensación inicial sobre qué ruta para dedicarse al diseño de dispositivos o estructuras y deberían idealmente ser el paso inicial antes de perseguir cualquier diseño detallado o análisis a través de la formulación diferencial.
El principio principal detrás de la formulación del volumen de control es cambiar los detalles de un sistema expuesto a un fluido por un diagrama de cuerpo libre simplificada definido por un imaginario cerrado superficie como del volumen de control. Este diagrama debe contener todas las fuerzas superficie y el cuerpo, el flujo neto de ímpetu linear a través de las fronteras del volumen de control y la tasa de cambio del ímpetu linear dentro del volumen de control. Este enfoque implica definir inteligentemente el volumen de control de manera que simplifican el análisis al mismo tiempo que captura los efectos dominantes en el sistema. Esta técnica se demostrará con un chorro de plano que inciden sobre una placa plana en diferentes ángulos. Utilizará el análisis del volumen control para estimar la carga aerodinámica en la placa y va a comparar nuestros resultados con las medidas reales de la fuerza resultante obtenida con un equilibrio aerodinámico.
Principles
Procedure
Results
Figure 3 shows a comparison between the normal load on the flat plate as measured directly from an aerodynamic balance and estimated from conservation of linear momentum. In general, the analysis of linear momentum captured the dominant tendency of direct measurements as the impingement angle changes. The discrepancies in these measurements varied non-monotonically with the impingement angle. For impingement angles in the range 
, and for 
, discrepancies are below 6%. They are higher for the other angles, but never higher than 12.5%. There appears to be a crossover around 
, in which the tendency of discrepancies invert: measurements exhibit higher normal loads than analysis of linear momentum for 
and lower for 
. These differences in the tendencies could be due to the fact that the analysis of linear momentum assumes inviscid, non-dissipative, changes in linear momentum, while direct measurements cannot avoid the effect of viscosity on the flow. For the range , the shear component becomes dominant and therefore turbulent boundary layer effects could be important. In this case, wall-normal velocity fluctuations due to turbulence might be responsible for the increase in the normal load. On the other hand, the axial velocity of the jet experiences a significant reduction in the range while it turns to become dominantly tangent to the wall. This effect is likely to let viscosity dissipate viscosity due to a reduction in the local values of Reynolds number, and that would result in reduced values of the normal load.
Table 2. Representative results.
| θ | F ̃_x(N) | F ̃_y (N) | F ̃_n (N) | F_n (N) | ε (%) |
| 90o | 15.257 | 9.034 | 15.257 | 16.773 | 9.9 |
| 85o | 15.151 | 9.831 | 15.950 | 16.709 | 4.8 |
| 82.5o | 15.035 | 10.231 | 16.242 | 16.630 | 2.4 |
| 80o | 15.929 | 10.498 | 17.510 | 16.518 | 5.7 |
| 75o | 14.248 | 10.453 | 16.468 | 16.202 | 1.6 |
| 70o | 13.518 | 11.405 | 16.604 | 15.762 | 5.1 |
| 67.5o | 13.100 | 11.294 | 16.425 | 15.496 | 5.7 |
| 65o | 12.771 | 11.579 | 16.468 | 15.202 | 7.7 |
| 60o | 11.881 | 11.863 | 16.221 | 14.526 | 10.5 |
| 50o | 9.746 | 11.241 | 14.691 | 12.849 | 12.5 |
| 40o | 6.357 | 9.444 | 11.320 | 10.782 | 4.8 |
Figure 3. Representative results. Load on plate as a result of impinging jet. Symbols represent: 
: direct load measurement; 
: estimation from conservation of linear momentum; 
: percent error between experimental measurements and theoretical estimation.
Applications and Summary
We demonstrated the application of control volume analysis of conservation of linear momentum to determine the forces exerted by a jet impinging on a flat plate. This analysis proved simple to apply, and gave a satisfactory bulk estimation of loads without requiring detailed knowledge of the flow pattern around the plate. Though there were some discrepancies (both in magnitude as well as tendency) due to the basic assumption of inviscid transformation of momentum, this technique offers a means of obtaining a quick estimation of system behavior without delving into a detailed study of fluid flow. Hence, this is a powerful tool for the engineering analyst to, for instance, predict the feasibility of developing a given engineering system with a minimal investment of time and resources. Once this first analysis is conducted to determine feasibility, the engineer can move into a more detailed flow analysis using, for example, computational fluid dynamics.
Control volume analysis of conservation of linear momentum is a powerful tool for fluids engineering. It finds application in a wide variety of problems to circumvent more involved methods such as differential analysis. A few instances of this analysis can be described:
Pelton turbine blade design: in general, a Pelton turbine blade should be designed to convert the highest amount of linear momentum into torque. This is achieved by determining the geometry of the blade that maximizes the change in the linear momentum of water jets. To this end, the typical result of control volume analysis is that the jet should be made to turn around on itself, that is, 180o. This is in general a technical challenge for a rotating device, but gives the analyst an initial guidance for a more detailed analysis using other tools.
Drag load on civil structures: one of the challenges of civil engineering is to design structures that stand the load of wind. In order to predict the effects of the wind on a real-size structure, it is possible to conduct experiments with a down-scaled model in wind or water tunnels. To this end, it is possible to use control volume analysis of conservation of linear momentum based on velocity measurements upstream and downstream of the model to determine the effective load on the prototype. This method both simplifies the experimental campaign and saves time, effort, and money in preparation for the construction of a real-scale structure.
References
- White, F. M. Fluid Mechanics, 7th ed., McGraw-Hill, 2009.
- Munson, B.R., D.F. Young, T.H. Okiishi. Fundamentals of Fluid Mechanics. 5th ed., Wiley, 2006.
- Buckingham, E. Note on contraction coefficients of jets of gas. Journal of Research,6:765-775, 1931.
- Lienhard V, J.H. and J.H. Lienhard IV. Velocity coefficients for free jets from sharp-edged orifices. ASME Journal of Fluids Engineering, 106:13-17, 1984.
Transcript
Control volume method is a powerful tool in fluid engineering, extensively used for the aerodynamic design of structures or devices. Force is developed when an object moves through a fluid. Forces exerted on bodies by a fluid flow are the result of changes in the linear momentum of the flow around them. In order to design a wind turbine blade, a boat sail, or an airplane wing, an engineer must be able to determine the dominant loads in a system. The toolbox of an engineering analyst contains methods to predict the feasibility of developing a given engineering system as well as complex methods for detailed structure calculation. This video will illustrate how to apply the control volume method to determine the aerodynamic load on a flat plate at different angles and demonstrate how loads can be estimated and measured in the laboratory.
Let’s consider a plane jet impinging on a flat inclined plate. You should be familiar with this example from our previous video. Now let’s take an arbitrary volume of interest around the structure named control volume, defined by an imaginary closed surface named control surface. The main principle behind control volume analysis is to replace the complex details of a system exposed to a fluid flow by a simplified free body diagram for the chosen volume. The forces acting on the system can be surface forces due to pressure or flow-induced shear. The forces acting on the system can also be body forces, for example the weight of the solids and fluids contained inside the control volume, or other forces induced by volumetric effects such as electromagnetic fields. The sum of the forces acting on the control volume equals the rate of change of the linear momentum inside the control volume and the net flux of linear momentum through the control surface, which also takes into account the speed of the control volume. This is the vector equation for the conservation of linear momentum. Now let’s come back to our example and apply the principles described earlier. First, let’s draw the control volume around the structure. The control volume must be chosen in ways that simplify the analysis and at the same time that capture the dominant effects on the system. Note that here the momentum flows into the control volume through port one and leaves through port two and port three. How can the equation of the momentum conservation be written for this particular configuration? Port one is placed at the location of the vena contracta where the fluid streamlines are parallel and the static pressure of the jet equals the atmospheric pressure. Assuming that ports two and three are located far enough from the impingement region, the same conditions are valid for these ports as well. Thus, the pressure is distributed homogeneously on the control surface and it is equal to atmospheric pressure. In consequence, the net pressure force acting on the control volume is zero. Since the control surface is perpendicular to the inlet and outlet flows, there is no shear load induced by the flow on the surface. The only term on the left-hand side of the equation is given by the reaction force of the plate to the transmission of the aerodynamic loading exerted by the jet on the plate. Assuming the jet flow is steady, there is no change of the momentum inside the control volume and thus the first term on the right-hand side of the equation vanishes. Since our control volume is fixed in space, the equation simplifies, showing that the reaction force to impingement equals the net flux of momentum through the control surface. The velocity vectors in our particular configuration of the control surface are aligned with the area vectors. In consequence, there is a negative influx at port one and outfluxes at ports two and three. The sum of these fluxes is the reaction force to impingement. Assuming that the velocity of the ports is approximately homogenous, the force equation simplifies further. Knowing the impingement angle theta, the resulting force can be decomposed into its normal component to the plate and its tangent component. Next, we find the normal and tangential components of the velocities at port one, port two, and respectively port three. We use these in the force equation in order to get the corresponding components of the force. The normal load on the plate is the most relevant from the structural point of view. It can also be expressed using the plate span and the width of the jet at the vena contracta. Knowing the contraction ratio between the jet exit width and the vena contracta and the dynamic pressure at the vena contracta, we obtain the final expression of the normal load on the plate estimated with the control volume analysis. In the next sections, we will measure the dominant forces exerted by an impinging jet on an inclined plate with an aerodynamic balance and then compare the measured load to the estimate based on the control volume analysis.
Before starting the experiment, check that the facility is not running. First, connect the positive port of the transducer to the pressure tap of the plenum. Leave the negative port of the pressure transducer open to sense the atmospheric pressure in the receiver. Zero the pressure transducer and record the value for the calibration constant. Set the jet exit width and measure the plate span. First, calibrate the aerodynamic balance to determine the lift conversion from volts to Newton and the drag conversion from volts to Newton. Next, record the volt to Newton conversion constants of the force aerodynamic balance device. Now record all the basic parameters of the experiment in a reference table. Next, set up the data acquisition system to capture a total of 500 samples at a rate of 100 Hertz corresponding to five seconds of data. Enter the volt to Newton conversion constants in the relevant fields of the data acquisition software. Finally, mount the impact plate on the force balance and adjust the device’s outputs to zero.
To start the data acquisition, first set the angle of the plate to 90 degrees and then turn the flow facility on. First, record the reading of the pressure transducer in volts. Use this quantity together with the calibration constant from the reference table in order to calculate the pressure difference between plenum and atmosphere. Now you are ready to measure the force with the force balance. To do this, use the data acquisition system to record force data. The data acquisition system will automatically use the conversion factors to determine force using the measurements in volts. Enter the results in a table. Turn the flow facility off and change the angle of the plate. Next, turn on the flow facility and repeat the force measurements for different angles. Record the data in a results table.
Calculate the normal force exerted on a flat plate by using the angle theta and the experimental values for the horizontal and the vertical components of the impingement force measured with the aerodynamic balance. Repeat the calculation for each angle theta and record the values in the results table. Using the parameters table and the measured values of the pressure difference between plenum and atmosphere, calculate the theoretical value of the normal impingement force on the plate. Repeat the calculation for each angle theta and record the values in the results table. Calculate the disagreement between the measured and theoretical values of the impingement force. Repeat the calculation for each angle theta and record the values in the results table.
Begin by plotting the load on the plate given by direct measurements with an aerodynamic balance as a function of the impingement angle theta. Place on the same graph the load calculated using the theoretical analysis with the control volume approach together with the percent error epsilon. Now compare the values directly measured with the values calculated with the control volume analysis for each load exerted on the plate at each angle theta. The discrepancies between the two methods vary non-monotonically with the angle theta and range between 2% and 12.5%. For angles smaller than and equal to 80 degrees, the control volume method underestimated the loads on the plate. While for angles higher than 80 degrees, this method gave values higher than the measured loads. The differences could be due to the fact that the control volume analysis assumes inviscid non-dissipative changes in linear momentum. While the direct measurements cannot avoid the effect of viscosity on the flow.
Control volume analysis of the conservation of linear momentum is widely used to predict the feasibility of developing a given engineering system before pursuing a detailed aerodynamic design of the structure or device. A Pelton blade is designed to convert the highest amount of linear momentum into torque. Control volume analysis has demonstrated that the blade geometry that maximizes the change in linear momentum of water jets is such that imposes a change of direction of 180 degrees in jet trajectory. In order to predict the effects of the wind on a real-sized structure, experiments can be conducted with a downscaled model in wind or water tunnels. Here the control volume analysis is used together with velocity measurements upstream and downstream of the model in order to determine the effective load of the prototype.
You’ve just watched Jove’s introduction to the control volume analysis of the conservation of linear momentum. You should now understand the basic principles of the method and how to apply them to estimate forces exerted by flow on a structure. You have also learned how to perform force measurements with an aerodynamic balance. Thanks for watching.