Dimensional Analysis - Kolmogorov Velocity Scale, VT in Homogeneous Turbulence - Coursework Example

FV2001 ASSIGNMENT BRIEF Name Institution Instructor Date 1. Dimensional analysis 1.1 Kolmogorov velocity scale, VT in homogeneous turbulence Turbulence is one of the significant phenomena in fluid dimensional analysis. It is commonly used and applied in a variety of fluid flow situations describe some of the most difficult situations. Dimensional analysis of the Kolmogorov velocity scale with regard to the homogeneous turbulence offers a structures understanding of the nature of turbulence. Turbulence is simple defined as the irregular or the stochastic change that is experienced in the parameters of a fluid (Cengel and Cimbala, 2010). The parameter may be such as fluid pressure, velocity or a process that involved chaotic fluid movement. Turbulence plays a significant role in several applications such hydraulics, aero-acoustics as well as transport of sediments. The mathematical description and dimensional analysis of turbulence is a very useful analytical tool that that is used in solving complex problems in applied science subjects (Raju, 2011). Dimensional analysis of Kolmogorov velocity scale, VT in homogeneous turbulence considers the following parameters: The coefficient of kinematic viscosity, v [m2/ s] Specific rate of dissipation, 𝟄 [J/Kgs] Density of the fluid, ƿ [Kg/m3] Velocity v = LT-1, Force F = MLT-2, Mass M = FL-1T2 Thus, the dimensional analysis of the kinematic viscosity yields L2T-1 The dimensional analysis of the fluid density yields ML-3 For kinematic coefficient of viscosity, it is the dynamic viscosity to the fluid density ratio and this is given by: Where v = velocity in [M3] µ = is the dynamic viscosity of fluid ƿ = Fluid density in [Kg/M3] Using the Reynolds number in the analysis of this concept, we have: Where L is typically the length The Reynolds number, Re = r (r𝟄r) 1/3/v Therefore, Kolmogorov velocity scale, VT in homogeneous turbulence becomes: 1.2 Demonstration of the Archimedes dimensionless number The Archimedes number, Ar is useful in the determination of the fluid motions resulting from differences in fluid density. The Archimedes dimensionless number is determined through gravitational force to viscous force ratio. The Archimedes dimensionless number has the form Where: g = Acceleration due to gravity (9.81 m/s²), ρe = the fluid density (Kg/M3), ρ = Body density (Kg/M3), µ = dynamic viscosity (Kg/ms), L = The characteristic body length (m) In the analysis of fluids with potential mixtures, the Archimedes dimensionless number becomes useful in providing parameters of the strength that is relative to both forced and free convections. In cases when The Archimedes number is more than one, it is clear that there is a dominance of the natural convection. This means that the dense bodies are sinking and the lighter ones are rising (Raju, 2011). However, when the Archimedes number is less than one, then there is the domination by the forced convection. When the transfer of heat results in the difference in density, the following expression is used in relation to the Archimedes principle, Where: ᵦ represents the coefficient of volumetric expansion coefficient T is temperature subscript 0 represents a point of reference within the body of fluid This usually takes place in the fluids that bring about temperature differences in fluid parts when they undergo heating. The performance of this, gives rise to another dimensionless number known as the Grashof nmber. This is because both the Grashof and Archimedes dimensionless numbers are equivalent and suitable in describing circumstances where therwe are differences with regard to density and heat transfer resulting in other differences. The relationship between the Archimedes dimensionless number and both Reynolds and Richardson dimensionless numbers is demonstrated by Where: Ar is the Archimedes dimensionless number Ri is the Richardson dimensionless number Re is the Reynolds dimensionless number 2. Classical Mechanics of Fluids 1.2 The Navier–Stokes equations The Navier-Stokes equation generally applies the Newton’s second law of motion, which is responsible for the momentum conservation. The momentum conservation equation applied in the general Navier-Stokes equation involves the conservation of both energy and mass. The general equation often takes an arbitrary form of the fluid portion. The Navier-Stokes equation is generally expressed as: Where: v is the velocity of flow, ρ is the density of fluid, p is the fluid pressure T is the total stress tensor component that is deviatory f is the force of the body per unit volume in the fluid The equation is often used with regard to material derivative which the Newton’s equation statement more apparent. The equation’s left side offers a description of the acceleration and it may include time convective or dependent effects. The equation’s right side is in effect a representation of the resultant body forces such as the divergence pressure, gravity and shear stress. Basic assumptions The Navier-Stokes equations are founded with regard to the basic assumptions, which include the assumption that the fluid with in the interest scale is a continuum. This means that it consists of continuous substances rather than discrete particles. Another assumption is the one that suggests that all the fields of interest such as energy, momentum, velocity, pressure and density among others are differentiable. The other one is the assumption that the derivation of the equations takes place from the primary principles of momentum, mass and energy conservation. The Navier-Stokes equations are as follows (Kundu, Cohen and Dowling, 2012): The physical meanings of the terms in the equations are as follows: Navier-Stokes equation with the force acting on the body, that is F = 0 Creping motion equation Navier-Stokes equation for low pressures The physical quantity for small viscous force in comparison to force of inertia The physical quantity equation for the absence of pressure force The terms in the Navier-Stokes equation that require turbulence modelling include the vector u and the scalar quantity p. This is because the two terms in the Navier-Stokes equation are unknowns and they are yielded from the continuity fluid flow equation. 2.2 Venturi meter The wide cross sectional area = 5.4 cm2 The narrow cross sectional area = 0.24 cm2 The level of water in the tank = 9m Appling the Bernoulli equation for fluids, we have: Where: ά It the turbulent flow correction v the average velocity of fluid at the cross sectional area of the pipes ζ is the pump efficiency Wp is the efficiency of pump fp is the friction correction factor Therefore, pressure drop between the narrow parts is given by: Assumptions used: Fluid density p = 1.29 Kg/m3 Coefficient of Discharge C = 0.98 Velocity at A, V= 0.199 m/s Volume Flow rate, Q= 0.104 l/s Mass Flow rate = 1.34 × 10-4 kg/s Pressure drop = 9.0 Pa 3. Heat Transfer, Thermochemistry and Fluid Dynamics of Combustion 3.1 Stoichiometric fuel-air ratio Stoichiometric fuel-air ratio for gasoline = 1/14.7 For lean mixture with fuel-air ratio 1/25, the use of the following formulas is applied: Where: 𝜆 is the stoichiometric ratio 𝜙 is the equivalence ratio F/A is the fuel-air ratio (F/A) s is the stoichiometric fuel-air ratio Assumption The stoichiometric air-fuel ratio for gasoline is 14.7 The oxidation reaction for gasoline fuel is: 𝜙 = (1/14.7) / (1/25) = 1.7 𝜆 = 1/1.7 = 0.588 Therefore mass fraction of gasoline in the mixture is: 2/(25 x 0.588) = 0.136 3.2 Normal flame velocity Equivalence ratio = 0.68 Mass fraction of diluent = 0.2 Diluent temperature = 179 o C Normal atmospheric pressure = 0.5 Pa Normal flame velocity =( (1/0.68) x 0.5) x 0.2 = 0.15m/s 4.0 Characteristics of Jet and Buoyant Flames and Fire Plumes 4.1 Main characteristics of Jet and Buoyant Flames and Fire Plumes Temperature The temperatures associated with the production and sustenance of fire flumes and jet flames is a key characteristic in the analysis of the flames. Temperatures determine the history times and lengths luminous flames occurring instantaneously (Modest, 2013). Volume The volume of the jet and the buoyant flames plays an important role in the investigation of features that are associated with luminous structures. The generation of volume for jet flames uses the idea of image processing. The volume in this case is a characteristic that allows for quantitative and qualitative comparison among flames (Lienhard and Lienhard, 2011). 4.2 Main characteristics of Fire Plumes and flows encountered in fire environments Speed As one of the main characteristics of the flows encountered in the fire environments and fire plumes, speed is a very significant feature of the flames. This because it influences the rate at which the plume flow changes direction in both outward and inward directed flows. Temperature The flame temperature in fire plume flow as well as the encountered fire environments is a function of the speed of airflow and the amount of fuel. Therefore, it also plays a role in determining the rate changes in flow direction (Modest, 2013). References Cengel, Y. A., & Cimbala, J. M. (2010). Fluid mechanics: fundamentals and applications. New Delhi, India, Tata McGraw Hill Education Private. Kundu, P. K., Cohen, I. M., & Dowling, D. R. (2012). Fluid mechanics. Waltham, MA, Academic Press. Lienhard, J. H., & Lienhard, J. H. (2011). A heat transfer textbook. Mineola, N.Y., Dover Publications. Modest, M. F. (2013). Radiative heat transfer. Oxford, Academic. Rajput, R. K. (1998). A textbook of fluid mechanics ("Fluid mechanics and Hydraulic machines"--Part I) in SI units. Ram Nagar, New Delhi, S. Chand. Raju, K. S. N. (2011). Fluid mechanics, heat transfer, and mass transfer chemical engineering practice. Hoboken, N.J., Wiley. Rathore, M. M., & Kapuno, R. R. (2011). Engineering heat transfer. Sudbury, Mass, Jones & Bartlett Learning. Read More