Fluid Dynamics: Velocity Profiles, Boundary Layers, and Turbulence
En flujo por encima de una placa plana dibuje el perfil de velocidades en la placa y el grosor de la capa límite y los regímenes que se observan.
Boundary Layer Thickness and Flow Regimes: The boundary layer thickness δ(x) increases with x, showing the growth of the boundary layer. Indicate the regions of laminar flow, transition, and turbulent flow along the length of the plate.
Deduzca el valor teórico del coeficiente de resistencia (drag coefficient) en flujo laminar sabiendo que el shear stress en la pared es: 
Write Bernoulli equation in the pressure energy form and head energy form
Which is the best turbine and why?
Demonstrate the formula and explain what Cc are and Cv
Diferencias entre agitación y mezclaAgitación: Acción mecánica que fuerza a un fluido a adquirir un movimiento circulatorio en el interior de un recipiente.Mezcla: Combinar dos o más materias distintas haciendo que las partículas queden unidas unas con otras, formando una cierta homogeneidad.Tipos de agitaciónAxial: El fluido va paralelo al eje de rotación. El agitador es una hélice y sirve para fluidos con bajas viscosidades.Radial: El fluido gira alrededor del eje de rotación. Sirve para fluidos con un amplio rango de viscosidades. Puede ser de palas verticales, ancla y turbinas.Usos agitadores de ancla: Está destinado para un amplio rango de viscosidades. Se puede utilizar en procesos de mezcla, dilución y homogeneización. Aplicaciones: Fusión de productos, mermeladas, control de temperatura, etc.¿Qué es la vorticidad? ¿Cómo se evita?
Vorticity is a measure of the local spinning motion of a fluid, defined as the curl of the velocity field ω⃗=∇⋅u⃗. It indicates the tendency of fluid elements to rotate around a point. High vorticity regions, such as vortices, show significant rotational motion, while zero vorticity indicates irrotational flow. To avoid or minimize vorticity, consider the following strategies: Streamlined Shapes: Design objects with streamlined shapes to reduce flow separation and vortex formation. Smooth Surfaces: Ensure surfaces in contact with the fluid are smooth to prevent boundary layer separation. Gradual Geometry Changes: Avoid sharp edges and sudden changes in geometry to maintain smooth flow. Boundary Layer Control: Use techniques like suction or blowing to control the boundary layer and prevent separation. Proper Flow Path Design: Design flow paths for smooth, uniform flow distribution. Active Flow Control: Implement methods such as synthetic jets or plasma actuators to manipulate the flow and reduce vorticity.
Demostración tubo de Venturi u1 = f(ΔP)
Buoyancy: described by the Archimedes principle, which says that a body immersed in a fluid experiences a vertical buoyant force equal to the weight of the fluid it displaces. A body can float only if it displaces its own weight in the fluid it floats. Centre of gravity and centroid must be aligned.Thermal expansion: low in liquids, very high in gases. Contact angle: is the one made by the surface’s tensions in the interfaces solid-vapour, solid-liquid and liquid- vapour. We can change the form of a bubble, for example of water, to change the wettability of the surface. If the angle is less than 90, the fluid wets.Non- slip condition: when a fluid enters in contact with a solid, the fluid acquires the same velocity as the surface. There is a gradient between a layer with null velocity and a layer with velocity, it will depend on the state of the fluid (if it is laminar or turbulent). In laminar conditions the fluid is ordered and smooth while in turbulent conditions it is chaotic and unsteady. In a laminar flow, each layer is solidary with the others while in turbulent flow layers are not solidary with the others, so we have different trajectories of the layers.Difference between gauge and absolute pressure: Pabs=Pgauge+PatmWhat is the correlation between the lift coefficient (CL) and the angle of attack (α)? The lift coefficient (CL) is a measure of the lift generated by a wing relative to the dynamic pressure of the airflow and the wing area. The angle of attack (α) is the angle between the chord line of the wing and the direction of the oncoming airflow.
Correlation: The lift coefficient (CL ) generally increases linearly with the angle of attack (α) for small angles. This is due to the increasing component of the airflow perpendicular to the wing surface, which generates more lift. This linear relationship is described by the formula:
C_L=C_L0+C_Lα⋅α . Nonlinearity at Higher Angles: As the angle of attack increases further, the relationship becomes nonlinear. Beyond a certain point, known as the critical angle of attack, the increase in lift coefficient slows down and eventually reverses due to flow separation and turbulence.
What happens to lift coefficient when the wing exceeds the critical angle of attack? The critical angle of attack is the angle beyond which the airflow can no longer smoothly adhere to the surface of the wing, leading to a condition known as stall.Flow Separation: When the wing exceeds the critical angle of attack, the smooth airflow over the wing’s upper surface separates, creating a turbulent wake. This separation dramatically reduces the lift generated by the wing.Sharp Decrease in Lift: As a result, the lift coefficient drops sharply, which can cause a sudden loss of lift. This drop is significant and can be catastrophic if not managed properly, as it leads to a decrease in the wing’s ability to support the aircraft.Stall Characteristics: The stall is characterized by a noticeable buffeting and loss of control, requiring corrective action to reduce the angle of attack and regain smooth airflow over the wing.
What is the order of magnitude of the drag coefficient (CD)? The order of magnitude of the drag coefficient depends on the shape and flow conditions around the object. For common shapes and typical flow conditions, the drag coefficient falls within the following ranges: Smooth Sphere: 0.1 to 0.5, depending on the Reynolds number. At low Reynolds numbers, it can be closer to 0.5, while at higher Reynolds numbers and during turbulent flow, it can drop to around 0.1. Flat Plate (Perpendicular to Flow): 1.0 to 2.0. Streamlined Bodies: 0.01-0.1. The drag coefficient for an airplane wing ranges from 0.01-0.05.
Lagrangian Description: Focuses on following the movement of individual fluid particles as they move through space and time. Each fluid particle has its own trajectory and behaviour is analysed individually. Useful for detailed studies of specific particles in the flow.
Eulerian Description: Focuses on analysing the behaviour of the fluid at fixed points in space as the fluid flows through these points. Describes the flow field in terms of variables like velocity, pressure, and density at each fixed point. More commonly used in fluid dynamics due to its mathematical and computational simplicity.
Radial Anchor: A radial anchor agitator is designed to scrape the bottom of the tank and prevent solid particles from settling. Its goal is to homogenize mixtures containing solids, ensuring they remain suspended and do not settle. The shape and movement of the agitator are designed to generate flow that covers the entire tank base and removes deposited solids.
Momentum in control volume: The local term of the momentum equation (continuity) varies when the mass within the control volume changes over time, occurring in unsteady (transient) flow. The advective term varies when the flow across the control volume surfaces changes, which happens when there are variations in flow velocity across these surfaces. In summary, the local term is associated with the accumulation or depletion of mass within the control volume, and the advective term is associated with the mass entering and exiting through the control volume boundaries.
Zero viscosity and normal stresses: When viscosity μ=0 shear stresses are eliminated, leaving only normal stresses. Normal stresses in a fluid with zero viscosity (ideal fluid) are equal to the pressure. The relationship between normal stresses and pressure is given by σij=−pδij where σij are the normal stresses and δij is the Kronecker delta.
Centrifugal pumps lowering efficiency: Incidence: Incorrect angle of attack of the impeller blades. Recirculation: Return flow within the pump. Friction losses: Viscous interactions within the pump.
Drag crisis: The drag crisis is a drastic reduction in the drag coefficient (CF) when the flow transitions from laminar to turbulent. This occurs due to the reduction of the boundary layer and decreased flow separation in turbulent flow.
Why the Navier-Stokes Equations Cannot Be Solved Analytically?: The Navier-Stokes equations are fundamental in describing the motion of fluid substances such as gases and liquids. Despite their importance, solving these equations analytically remains an unsolved challenge due to their inherent complexity. One primary difficulty lies in their nonlinearity, particularly the convective term (u⋅∇)u, which causes small changes in initial conditions to result in large and unpredictable changes in the solution. This nonlinear behaviour makes finding exact solutions exceedingly difficult. Additionally, the Navier-Stokes equations are coupled, meaning they simultaneously describe multiple interdependent physical phenomena, including momentum, mass, and energy conservation. This coupling requires solving for multiple variables that affect each other in intricate ways, further complicating the quest for analytical solutions.
Moreover, the equations are often three-dimensional in real-world scenarios, adding to their complexity. Solving three-dimensional partial differential equations analytically is typically infeasible due to the vast number of variables and interactions involved. The presence of turbulence, characterized by chaotic and seemingly random fluid motion, adds another layer of difficulty. Turbulent flows are particularly challenging to model and predict due to their complex and unpredictable nature.
Boundary and initial conditions also play a crucial role, as solutions to the Navier-Stokes equations are highly sensitive to these parameters. Specifying accurate boundary conditions for practical problems is often difficult, leading to further complications in finding exact solutions. Finally, one of the major unsolved problems in mathematics is proving the existence and smoothness of solutions to the Navier-Stokes equations in three dimensions. It’s not known whether smooth, globally defined solutions always exist or if singularities (points where the solution becomes infinite) can develop in finite time. This fundamental uncertainty underscores the difficulty of solving these equations analytically.
Simplifications to Work with the Navier-Stokes Equations To make the Navier-Stokes equations more tractable, several simplifications are often applied depending on the specific problem and context. These simplifications help to reduce the complexity of the equations and make analytical or numerical solutions more feasible. Incompressible Flow: Assuming the fluid density is constant simplifies the continuity equation. This assumption is valid for liquids and low-speed gas flows where density changes are negligible. Steady-State Flow: In steady-state flow, all time-dependent terms are set to zero. This assumption simplifies the equations significantly, as it removes the need to account for changes over time. Laminar Flow: Assuming laminar flow, where fluid flows in parallel layers without disruption, simplifies the equations. This is in contrast to turbulent flow, which is much more complex to model. Neglecting Viscosity: For inviscid flow (ideal fluids), the viscosity term can be neglected. This leads to the Euler equations, which are simpler to solve than the full Navier-Stokes equations. Symmetry and Simplified Geometry: Assuming symmetry and working with simplified geometries (e.g., flow in a pipe or between parallel plates) can reduce the equations to a more manageable form. Symmetry reduces the number of spatial dimensions that need to be considered. Boundary Layer Approximation: In high Reynolds number flows, the boundary layer approximation can be used. This involves simplifying the equations within the thin region near the boundary where viscous effects are significant, while treating the rest of the flow as inviscid. Low Reynolds Number: Where viscous forces dominate inertial forces, the equations can be simplified to the Stokes flow equations. This simplification is particularly useful for modelling slow, viscous flows.
By applying these simplifications, it becomes possible to derive analytical solutions for specific, idealized cases of fluid flow. These simplified models provide valuable insights and approximate solutions that are useful in engineering and scientific applications. However, for more complex and realistic scenarios, numerical methods are often employed to solve the Navier-Stokes equations.