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Unformatted text preview: Chapter 3 Integral Boundary Layer Equations for ThreeDimensional Flows 3.1 Deﬁnitions The threedimensional integral boundary layer equations derived using a Cartesian coordi nate system are: Giff; = K. (oi/2“, — 0;”) (31)
%(peq30;) + %(p¢q302) + peg??? it: + p.436? 6;: = 20 (32)
13—29430”) + % (9.413022) + peqeﬁ; % + 9.41.52 23% = 72w (33)
% (“.130”) + £04139“) + Peqeli; % + new; % = Tm (34) Equation 3.1 is the twodimensional 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 3.23.4 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 22 coordinate system is an arbitrary, local, 2D surface
coordinate system in a plane tangent to the 3D 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 streamwisecrossﬂow coordinate system (called the 12 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 :cmomentum deﬁcit due to ﬂuxes in
the zdirection ((243033) and zmomentum deﬁcit due to ﬂuxes in the zdirection @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 zmomentum deﬁcit. 3.2 Boundary Layer Domain The threedimensional 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 threedimensional free transition will not be 34 ...
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 Fall '03
 MarkDrela
 Dynamics, Energy, Force, Kinetic Energy, Aerodynamics, Shear Stress, Boundary Layer Equations

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