three_dimensional_flows

# three_dimensional_flows - Chapter 3 Integral Boundary Layer...

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Unformatted text preview: Chapter 3 Integral Boundary Layer Equations for Three-Dimensional Flows 3.1 Deﬁnitions The three-dimensional integral boundary layer equations derived using a Cartesian coordi- nate system are: Giff; = K. (oi/2“, — 0;”) (3-1) %(peq30;) + %(p¢q302) + peg??? it: + p.436? 6;: = 20 (3-2) 13—29430”) + % (9.413022) + peqeﬁ; % + 9.41.52 23% = 72w (3-3) % (“.130”) + £04139“) + Peqeli; % + new; % = Tm (3-4) Equation 3.1 is the two-dimensional turbulent shear stress lag equation, 3.2 is the kinetic energy equation, and 3.3 and 3.4 are the a: and z momentum equations. The origin of equation 3.1 is given in reference [21] and the derivation of equations 3.2-3.4 may be found in [38]. K c is an empirical constant set to 5.6. f is the lag direction which is taken to be the chordwise direction, and the 2-2 coordinate system is an arbitrary, local, 2-D surface coordinate system in a plane tangent to the 3-D BL surface. The main assumptions in these equations are that pressure is constant in the normal direction through the thickness of the boundary layer, and that only diffusion normal to the wall is signiﬁcant. no In Equations 3.1 to 3.4, 6*’s denote diSplacement thicknesses, 9’s signify momentum thicknesses, 0"s represent energy thicknesses, and 6"’s are density thicknesses. In addition, (711/2 is the shear stress coefﬁcient, 03/25., is the equilibrium shear stress coefﬁcient, ‘r’s are wall shear stresses, and D is the turbulent dissipation fumtion. Section §B.1 provides the deﬁnitions of the thicknesses for a streamwise-crossﬂow coordinate system (called the 1-2 coordinate system). These integral boundary layer equations have a physical interpretation. Equation 3.2 may be thought of as a divergence of kinetic energy deﬁcit (the peqSO; and p439; terms) balanced by mechanical work deﬁcit (peq36;‘%, peq36;‘%‘£z‘) and dissipation The mechanical work terms are the products of pressure forces and densities; the pressures may be recogniZed using Euler’s Relation dp = —pqdq; and the densities may be seen by the deﬁnitions of qe6;‘,qe6;‘. Equation 3.3 contains the :c-momentum deﬁcit due to ﬂuxes in the z-direction ((243033) and z-momentum deﬁcit due to ﬂuxes in the z-direction @430“) balanced by pressure gradient term (peqc6;% + p,q,6;§a—“z‘) and wall shear stresses (TN). The pressure gradient term is Seen once again using Euler’s Relation. Equation 3.4 is a similar expression for z-momentum deﬁcit. 3.2 Boundary Layer Domain The three-dimensional boundary layer equations are a set of hyperbolic partial differential equations [59, 11]. Consequently, these equations require an initial or starting solution, and boundary conditions which depend on the mathematical characteristics entering or leaving the domain. A typical boundary layer domain is depicted in Figure 3.2. The starting solution for Equations 3.2 to 3.4 is speciﬁed at the attachment line near the wing leading edge, and the starting solution for Equation 3.1 is speciﬁed at the transition line. Boundary conditions are applied along the symmetry line, and the wingtip and waketip boundaries. The main purpose of this research is to implement Fully Simultaneous coupling in three dimensions which is a difﬁcult problem in itself. Therefore more complicated issues such as comer ﬂows (at wing / body junctures) and three-dimensional free transition will not be 34 ...
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three_dimensional_flows - Chapter 3 Integral Boundary Layer...

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