Threedimensional (3D) viscous counterflows and wall stagnation flows are analyzed with differing normal strain rates in each of the three directions. Reduction of the equations to a similar form is obtained allowing for variations in density due to temperature and composition, heat conduction, and, for the counterflow, mass diffusion and the presence of a flame. Solutions to the NavierStokes equations are obtained without the boundarylayer approximation. For the steady and unsteady incompressible counterflows, analytical solutions are obtained for the flow field and the scalar fields subject to heat and mass transfer. In steady, variabledensity configurations, a set of ordinary differential equations (ODEs) governs the two transverse velocity and the axial velocity profiles as well as the scalarfield variables. Diffusion rates for mass, momentum, and energy depend on the two normal strain rates parallel to the counterflow interface or the wall and thereby not merely on the sum of those two strain rates. For thin diffusion flames, the location, burning rate, and peak temperature are readily obtained. Solutions for planar flows and axisymmetric flows are obtained as limits here. Results for the velocity and scalar fields are found for a full range of the distribution of normal strain rates between the two transverse directions, various Prandtl number values, and various ambient (or wall) temperatures. For counterflows with flames and stagnation layers with hot walls, velocity overshoots are seen in the viscous layer, yielding an important correction of theories based on a constantdensity assumption.
NOMENCLATURE
 c_{p}

specific heat under constant pressure [J/(K kg)]
 D

mass diffusivity (m^{2}/s)
 H

specific enthalpy defined before Eq. (6) (J/kg)
 $ H \u0303 =h+ ( u 2 + v 2 + w 2 ) /2\u2248h+ v 2 /2$

alternative specific enthalpy (J/kg)
 h

specific enthalpy (J/kg)
 h_{f,m}

heat of formation for species (mJ/kg)
 Le=λ/(c_{p}ρD)

Lewis number
 M

Mach number
 N

number of species
 p

pressure (N/m^{2})
 Pr

Prandtl number
 R

specific gas constant [J/(K kg)]
 R_{u}

universal gas constant [J/(K mole)]
 S_{1}

normal strain rate in the x direction (s^{−1})
 S_{2}

normal strain rate in the z direction (s^{−1})
 t

time (s)
 u, v, $w$

velocity components (m/s)
 x, y, z

Cartesian coordinate (m)
 Y_{m}

mixture fraction
 Z

mass fraction of species
 α, β

conserved scalars
 η

density weighted coordinate (kg/m^{2})
 λ

thermal conductivity [J/(sm^{2})]
 ρ

density (kg/m^{3})
 $ \omega \u0307 m $

reaction rate for species (s^{−1})
 τ_{ij}

viscous stress tensor (N/m^{2})
Superscripts
Subscripts
I. INTRODUCTION
Here, we treat a threedimensional (3D) counterflow with differing strain rates in each direction. It can address counterflows in nonsymmetric fields or wall stagnation flows for nonsymmetric aerodynamic bodies. Two counterflowing streams might be constrained to have greater acceleration in one transverse direction than in the other. The aerodynamic body at the stagnation point might have differing radii of curvature in two mutually orthogonal planes which are each orthogonal to the tangent plane at that stagnation point.
There is growing interest in understanding the laminar mixing and combustion that commonly occurs within turbulent eddies. These laminar subdomains experience significant strain. Some important work has been done here but typically in twodimensions or axisymmetry with a constantdensity approximation. See the work of Linan,^{1} Marble,^{2} Karagozian and Marble,^{3} Cetegen and Sirignano,^{4,5} and Peters.^{6} Karagozian and Marble did examine a threedimensional strainedflow configuration where flow moved radially inward and jetted axially outward; in addition, a vortex had the same axis as the flow. The vortex caused the diffusion flame sheet to wrap around the axis. Recently, Rajamanickam et al.^{7} have provided an interesting threedimensional tripleflame analysis; the imposed strain however is limited to two dimensions. There is a strong need to study mixing and combustion in counterflows where the imposed strain is threedimensional. Note that Nguyen et al.^{8} and Nguyen and Sirignano^{9} have recently shown the importance of strained triple flames in practical combustors.
The “counterflow” label has been given to several types of distinct flows. One type involves two parallel, adjacent flows in opposite directions and possibly in different conduits. These flows are common in heat exchangers.^{10} Another type of counterflow is actually a recirculating flow caused by an imposed swirl.^{11} A third kind involves two mutually penetrating flows such as that found with a superfluid and normal fluid moving in opposite directions through the same volume.^{12} The references above are merely samples; a thorough review of those types of counterflows is not intended. Here, we focus only on a fourth kind where two streams move toward each other in the y direction with velocities of opposite signs but then turn toward the x and z directions before they make contact with each other at an interface in a plane normal to the y direction. $v$, the y component of velocity, becomes zero at this interface.
There is a wellestablished literature for steadystate viscous twodimensional and axisymmetric counterflows and stagnationpoint flows.^{13} Strahle^{14} examined the axisymmetric, unsteady stagnation point, including the presence of heat and mass transport and a diffusion flame. However, threedimensional viscous flows of these types have not received very much fundamental treatment. Howarth^{15} has treated the steady, 3D wall stagnationpoint flow for an incompressible fluid. Heat transfer was not considered there. Here, generalizations for variabledensity viscous counterflows and wallstagnation flows are presented for several basic configurations. Density variations through the flow due to variations in temperature, pressure, and/or composition are considered. Heat transfer caused by different temperature values between the two opposing streams or between the incoming stream and a wall is examined. Also, mass transfer for the counterflow caused by differing compositions of the two opposing streams is considered. A counterflow with a diffusion flame is also considered. Still, there are some similarities between the approach here and the portions of Howarth’s analysis.
Substantial work has occurred for planar and axisymmetric counterflow diffusion flames with singlephase flows at moderate pressures.^{1,6,16–19} The axisymmetric counterflow configuration has also been studied for spray combustion^{20,21} and at supercritical pressures.^{22} Here, an extension will be made for a threedimensional counterflow at moderate pressures and a gaseous phase.
The prior counterflow and stagnationpoint studies (cited above) address typically twodimensional configurations (planar or axisymmetric). As an exception, Howarth addresses threedimensional, incompressible wall flow. They obtain a similar solution whereby a system of ordinary differential equations (ODEs) with only one spatial coordinate remaining as the independent variable; the description of dependence on the other spatial coordinate(s) is given in algebraic terms. Our work here presents similar solutions for threedimensional flows where again only one coordinate appears as the independent variable. The onedimensional appearance of the final governing equations is not a model but still is an exact solution of a multidimensional configuration.
Aerodynamic studies of the shape of the interface between two counterflows have also been performed.^{23,24} Here, a freestream velocity and a radius of curvature of a counterflowinterface or a wall radius of curvature will not be specified. Rather, the order of magnitude of that ratio of velocity to radius is implied through the prescribed strain rate. The incoming counterflow streams will have a potentialflow character.
Here, we address a set of threedimensional counterflows and stagnationpoint flows, steady and unsteady, with and without combustion, and with and without constant density and properties. These various problems have differences but are not totally independent.
II. ANALYSIS
Consider, for example, a counterflow from both the negative ydirection and positive ydirection with outflow in the zdirections. Inflow or outflow can occur in the xdirection with the interface surface along y = 0 and a stagnation point at x = y = z = 0. The velocity $ u \u2192 $ has the components u, $v$, and $w$ in the x, y, and z directions, respectively. The flow must have a negative normal strain rate in at least one direction; the ydirection has arbitrarily been chosen here to be the direction with the only negative strain rate. The zdirection has the larger positive (or equal) strain rate, while the xdirection has a smaller (or equal) normal strain rate. If the approaching streams have the same pressure at a distance from the interface and its viscous layer, we expect that in a frame of reference attached to the interface, momentum balance for steady flow yields $ \rho \u2212 \u221e v \u2212 \u221e 2 = \rho \u221e v \u221e 2 $. The incoming flow is in the direction of decreasing ymagnitude. The normal strain rates in the x and zdirections are S_{1} = ∂u/∂x and S_{2} = ∂$w$/∂z, respectively. Here, S_{1} + S_{2} > 0 and S_{2} > 0. S_{1} can be positive or negative. These normal strain rates for these two opposing streams will match at the interface since the velocity components are continuous there. We are able to find a solution in the stagnation point region where these two strain rates apply throughout the viscous layer and do not vary with y. We consider only S_{1} ≥ 0, which is a classical counterflow with the interface at y = 0. If S_{1} < 0, there would be sink inflow in the x and ydirections with outjetting in the zdirection. This case is not addressed by our similarsolution approach. The ydirected inflowing streams in all cases bring together fluids of differing temperature and/or composition; so, heat and mass diffusion are in the ydirection.
The two streams need not have the same upstream values for velocity $v$, temperature T, enthalpy h, density ρ, or composition reflected through mass fraction Y_{m} for chemical species m. Pressure p will be given the same upstream values for the two streams. Fickian mass diffusion and Fourier heat conduction are considered so that all fluid properties are continuous across the interface. Some consideration is given to reacting flows. Radiation and gravity are neglected. The formulation of the partial differential equations below will also apply for a stagnating flow approaching a wall at y = 0.
With the addition of terms accounting for chemical rates for exothermic reactions, diffusion flames can be addressed by the analysis here. With a variation in the boundary conditions, the analysis will apply to wall stagnationpoint flows. There are several studies of planar and axisymmetric counterflow diffusion flames with account given to variable density;^{17–19} no prior works on 3D strained diffusion flames are known.
The nondimensional forms of the above equations remain identical to the above forms if we choose certain reference values for normalization. In the remainder of this article, the nondimensional forms of the above equations will be considered. The superscript * will be used to designate a dimensional property. The variables $ u i * , t * , x i * , \rho * , h * , p * $, and $ \omega \u0307 m * $ and properties $ \mu * , \lambda * / c p * $, and D* are normalized, respectively, by $ [ ( S 1 * + S 2 * ) \mu \u221e * / \rho \u221e * ] 1 / 2 , ( S 1 * + S 2 * ) \u2212 1 , [ \mu \u221e * / ( \rho \u221e * ( S 1 * + S 2 * ) ) ] 1 / 2 , \rho \u221e * , ( S 1 * + S 2 * ) \mu \u221e * / \rho \u221e * , ( S 1 * + S 2 * ) \mu \u221e * , ( S 1 * + S 2 * ) , \mu \u221e * , \mu \u221e * $, and $ \mu \u221e * / \rho \u221e * $. It is understood that for unsteady flow, the reference values for strain rates and farstream variables and properties used for normalization will be constants; for example, averages might be taken for fluctuating conditions. Note that the reference length $ [ \mu \u221e * / ( \rho \u221e * ( S 1 * + S 2 * ) ) ] 1 / 2 $ is the estimate for the magnitude of the viscouslayer thickness.
The stagnation point either in the steady counterflow or in the steady flow against a wall will be taken as the origin x = y = z = 0. Along the line x = z = 0 normal to the interface or wall, we can expect the first derivatives of $v$, ρ, h, T, and Y_{m} with respect to either x or z to be zerovalued. For unsteady cases, only symmetric situations will be considered so that the stagnation point remains at the origin and the wall or interface remains at y = 0. The velocity components u and $w$ will be odd functions of x and z, respectively, going through zero and changing the sign at that line. Consequently, upon neglect of terms of O(x^{2}) and O(z^{2}), the variables $v$, ρ, h, T, and Y_{m} can be considered to be functions only of t and y. For steady flow, the densityweighted Illingworth transformation of y can be used to replace y with $\eta \u2261 \u222b 0 y \rho ( y \u2032 ) d y \u2032 $. Neglect of the same order of terms implies that u = S_{1}x(df_{1}/dη) and $w$ = S_{2}z(df_{2}/dη). Note that u is independent of z and $w$ is independent of x in this case where no shear strain is imposed on the incoming stream(s). At the edge of the viscous layer at large positive η, df_{1}/dη → 1, df_{2}/dη → 1, f_{1} → η, and f_{2} → η. We define ()′ ≡ d()/dη. Note that other transformations of the y coordinate can be made, e.g., weighting by transport properties^{17,18} rather than density.
III. 3D COUNTERFLOW
The values of $v$ do not asymptote to a constant at +∞ or −∞; we therefore will take the boundary values for $v$ at y* positions whose magnitudes are severalfold the expected viscouslayer thickness, $ [ \mu \u221e * / ( \rho \u221e * ( S 1 * + S 2 * ) ) ] 1 / 2 $. So, +∞ is approximated by a value of y ≫ 1 or η ≫ 1, while −∞ is approximated by a value of y ≫ 1 or η ≫ ρ_{−∞}. Variable density due to temperature variation will be addressed. However, compressibility effects within the viscous layer will be assumed to be negligible. The magnitude of the dimensional velocity at the layer edge is $ [ ( S 1 * + S 2 * ) \mu \u221e * / \rho \u221e * ] 1 / 2 $; thus, the kinetic energy per mass based on that velocity magnitude will be assumed to be small compared to the ambient enthalpy values.
A. Steady incompressible counterflow
B. Unsteady incompressible counterflow
Solutions for the unsteady incompressible energy and species Eqs. (17) and (18) are also possible in the unsteady state with constant λ/(ρc_{p}) and Le = 1.
The incompressible steady or unsteady counterflow does experience diffusion of heat and/or mass from one flow toward the other when the two opposing, incoming streams have differing temperature and/or composition. Since the two incoming streams have the same xmomentum, ymomentum magnitude, and zmomentum, there is no diffusion of momentum from one stream toward the other. Here, although there is a nonzero $dv$/dy value through the viscous layer, it is not caused by diffusion but rather by the pressure gradient, i.e., it is independent of μ and the same velocity solutions for u, $v$, and $w$ occur if μ = 0. In the variable density case, the variation of density and/or properties through the interface layer does result in diffusion of momentum from one stream toward another.
In both the steady and unsteady incompressible counterflows, all three velocity components are linear in the spatial coordinate variables. Thereby, second derivatives of velocity are everywhere zero, yielding zero viscous force everywhere. However, the constant first derivatives result in a uniform viscous dissipation rate over the full space, amounting to a uniform heat source which is being neglected here. In particular, $ ( \u2202 u i / \u2202 x j ) \tau i j =4\mu ( S 1 2 + S 1 S 2 + S 2 2 ) $. This same value of the viscous dissipation rate will occur asymptotically outside the viscous layer for the variabledensity counterflow and the constantdensity and variable density wallstagnation flows. Thus, a heatsource term that would appear in Eqs. (17) and (21) is ignored here, assuming its effect is minor for our situations where kinetic energy is small compared to thermal energy. Although counterflows and wallstagnation flows are posed theoretically as extending to infinity, the curvature of the counterflow interface or wall will limit the domain size. Thereby, this small heat source does not in practice extend over an infinite domain and its global impact remains small. This conclusion will apply to the variabledensity flows as well.
For the incompressible counterflow case, the results here for the velocity fields can be extended to the case where two streams are incoming (say, in the y and x directions) and only one (say in the z direction) is outgoing; i.e., S_{1} < 0 but S_{1} + S_{2} > 0. The reason is that u and $w$ do not vary with y; so, a plausible incoming stream from the xdirection occurs. (Of course, the flow in the z and y directions could be incoming with outgoing flow in the x direction.) However, inflow conditions on one stream for the scalar variables would have to be too contrived to make those scalar results useful in that case. That is, the scalar variables for incoming flow in the x (or z) direction would have to satisfy specifically the y dependence given by Eq. (19) or (25).
C. Variabledensity counterflow
The variable density and viscosity case requires some couplings with Eqs. (4) and (5) and with an equation of state and fluidproperty laws which affect ρ and μ.
An exact solution of the variabledensity NavierStokes equation has been obtained subject to determination of ρ and μ through solutions of the energy and species equations as discussed below. There has been no need for use of a boundarylayer approximation. Thus, the solution here is the natural solution, subject to neglect of terms of O(x^{2}) and O(z^{2}). Unlike the incompressible counterflow, a viscous layer exists with the three normal strains and normal viscous stresses varying through the layer due to varying density and viscosity. Shear strain also exists.
In the above analysis, no boundarylayer approximation was used. It remains to use thermodynamic relations to substitute for ρ and μ in terms of $h= H \u0303 \u2212 v 2 /2$ and p.
Equation (11) indicates a dependence of the heat and mass transport on f ≡ S_{1}f_{1} + S_{2}f_{2}. Manipulation of the first two equations of (11) leads to an ODE for f with S_{1}S_{2} and $ S 1 S 2 f 1 \u2032 f 2 \u2032 $ as parameters, clearly indicating that generally f will have a dependence on S_{1}S_{2}. Thus, the behavior for the counterflow can vary from the planar value of S_{1} = 1, S_{2} = 0 (or vice versa) or from the axisymmetric case S_{1} = S_{2} = 1/2. This clearly shows that distinctions must be made amongst the various possibilities for threedimensional strain fields as S_{1}S_{2} varies between large negative numbers and 1/4. An exception will be the incompressible case with constant properties where the S_{1}S_{2} terms cancel in the equation for f.
The effects of the normalized strain rates (S_{1} and S_{2} = 1 − S_{1}) are shown in Figs. 2 and 3. A simple inspection of the governing ODEs leads to the conclusion that the values for $ f 1 , f 1 \u2032 , f 2 , f 2 \u2032 ,u/x$, and $w$/z can be interchanged with the values for $ f 2 , f 2 \u2032 , f 1 , f 1 \u2032 ,w/z$, and u/x, when S_{1} and S_{2} are replaced by 1 − S_{1} and 1 − S_{2}, respectively. Thus, only values for S_{1} ≤ 0.5, S_{2} ≥ 0.5 are reported. S_{1} = 0, S_{2} = 1.0 is the planar (twodimensional) case, while S_{1} = 0.5 = S_{2} is the axisymmetric case. Note that for S_{1} > 1 or S_{2} > 1 (which imply S_{2} < 0 or S_{1} < 0, respectively), there would be incoming streams from two directions. One incoming stream would have a prescribed velocity profile in the viscous layer determined as a local exact solution to the NavierStokes conditional on matching the profile determined by upstream conditions for the flow in that direction; this situation is too highly contrived and is not considered here. Thus, S_{1} and S_{2} are always each nonnegative and bounded above by the unity value in our considerations here. The figures show results for three strain rates: S_{1} = 0 (planar case), S_{1} = 0.25 (3D strain), and S_{1} = 0.5 (axisymmetric case).
The results for the threedimensional case, exemplified here by S_{1} = 0.25, lie between the results for the planar case (S_{1} = 0) and the axisymmetric case (S_{1} = 0.5). The strain rate has no noticeable affect on $ f 2 \u2032 $ which applies to the direction with the greater positive normal strain rate and a slight effect on $ f 1 \u2032 $ for the lower strained direction. Consistently, there is little effect of strain rate on f_{1} and f_{2}. The term $ S 1,2 [ h \u0303 \u2212 ( \u2009 f 1,2 \u2032 ) 2 ] $ in Eq. (31) is too small to create a substantial effect; it is found to be typically two orders of magnitude smaller than the $ f 1,2 f 1,2 \u2033 $ term. The nondimensional enthalpy and the nondimensional velocity component $v$ in the counterflow direction show no significant effect from the distribution of the transverse normal strain rate. The dimensional velocity component in the counterflow direction would scale with the square root of the dimensional strain rate. The values of u/x and $w$/z do depend strongly on the nondimensional strain rate in their respective directions with a direct proportion. The dimensional values u*/x* and $w$*/z* would then scale with the dimensional strain rate in the ydirection. The influence of Pr is shown in Fig. 4. As expected, an increase in the value of Pr causes a larger gradient for the enthalpy variation. Interestingly, a similar steepening is caused for $ f 1 \u2032 , f 2 \u2032 ,u/x$, and $w$/z. Little effect is seen for the $v$ component of velocity. The variation of 1/ρ_{−∞} = h/h_{∞} is described in Fig. 5. Since the boundary conditions for h, f_{1}, and f_{2} at η = −∞ are substantially affected, there are strong consequences for the solutions, especially for negative η. The consequences are focused there since f_{1}(0) = f_{2}(0) = 0 and $ f 1 \u2032 ( \u221e ) = f 2 \u2032 ( \u221e ) =1$ are maintained for all the cases here. Note that the density weighting for the variable η makes the inflection stronger as the solution varies from the higherdensity, negativeη range to the lowerdensity, positiveη range. See, for example, the curves for $v$ vs η. As shown in Fig. 2, ρ$v$ = S_{1}f_{1} + S_{2}f_{2} is not generally linear in η. Hence, the solution for scalars obtained from diffusiveadvective equations such as Eq. (32) will not yield error functions as commonly found with a constantdensity assumption.^{6} Equation (19) gives such an example of the constantdensity assumption.
In related fashion to the incompressible counterflow, the variabledensity results here for the velocity fields can in principle be extended to the case where two streams are incoming (say, in the y and x directions) and only one (say, in the z direction) is outgoing; i.e., S_{1} < 0 but S_{1} + S_{2} > 0. However, now u and $w$ do vary with y; so, the inflow velocity conditions as well as the scalar variable conditions are not plausible.
IV. DIFFUSION FLAME WITH 3D STRAIN
In the above energy equation, we have retained the ycomponent of kinetic energy per unit mass in H. However, an order of magnitude analysis indicates that it can be neglected in practical situations. Specifically, for a practical fluid at moderate temperature, a value of $v$*^{2} of O(10^{3} m/s) or greater is required for the kinetic energy to be at least one percent of the sensible enthalpy. Then, the viscouslayer thickness $ \delta * =O ( 1 0 3 / 2 / ( S 1 * + S 2 * ) ) $ and also $ \delta * 2 =O ( \mu * / [ \rho * ( S 1 * + S 2 * ) ] ) $ from the advectivediffusive balance of the governing equations. With practical values of μ* and ρ*, the strain rate $ S 1 * + S 2 * $ will reach O(10^{7}/s) or greater. This is much too large to allow chemical reaction and to hold a flame. So, here in this subsection, we redefine $H\u2261h+ \Sigma m = 1 N Y m h f , m $ neglecting the kinetic energy. Then, no assumption about Pr is made.
For the case of a simple diffusion flame with Y_{F} and Y_{O} representing the mass fractions of fuel and oxidizer, the conserved scalar is α = Y_{F} − νY_{O}. The far incoming stream on one side (positive y and η) has Y_{F} = 1, while the other far incoming stream (negative y and η) has Y_{O} = 1. For a rapid reaction rate, a thin flame occurs and can be assumed to have zero thickness. The thin flame is positioned at the η value where α = 0. At the η (or y) position where α = 0, the magnitude of the local value of ρ$v$α − (ρμ/Pr)dα/dη gives the burning rate (mass flux per area). It is the sum of transport into the thin flame of fuel vapor by advection and diffusion.
In Eq. (34), realize that the nondimensional reaction rate ω_{i} becomes very small when the dimensional strain rate is much greater than the dimensional reaction rate. This can cause extinction of a flame due to the stretching effect. Extinction will not be examined here.
Figures 6–8 give the computational results for Q = 10, ν = 0.25, and various values of S_{1}, S_{2}, Pr, and ρ_{−∞}. The solutions for the conserved scalars α and β and the velocity component $v$ are monotonic. However, the enthalpy $ h \u0303 $ peaks at the reaction zone and all of the velocity components u, v, and $w$ have overshoots in the lowdensity region around the reaction zone. Since the BurkeSchumann limit of the infinite reaction rate is employed, the reaction zone has zero thickness resulting in a cusped shape with a discontinuity of the first derivative for $ h \u0303 $. The rate of strain has modest affect on the scalar fields and $v$. Significant influence is seen on the u and $w$ velocity components which each increases with normal strain values for their respective direction. Unlike the nonreacting counterflow, the term $ S 1,2 [ h \u0303 \u2212 ( \u2009 f 1,2 \u2032 ) 2 ] $ in Eq. (31) is large for the reacting flow; it is found to be of the same order of magnitude as the $ f 1,2 f 1,2 \u2033 $ term. Again, the results are readily extended since the values for u/x and $w$/z can be interchanged with the values for $w$/z and u/x when S_{1} and S_{2} are replaced by 1 − S_{1} and 1 − S_{2}, respectively.
The results for the scalar properties have increasing gradients as Pr increases. In Fig. 7, the domain of high temperature and low density is narrowed in η space as Pr increases because scalar gradients increase. Thereby, densities in the lowdensity region increase causing the domain width in y space to narrow even more as Pr increases. As a consequence of the increase in density, the magnitude of the velocity overshoot is decreased as Pr increases. Figure 8 shows that an increase in ambient temperature leads, as expected, to higher peak values of temperature and enthalpy and to a greater velocity overshoot for all components.
Bilger^{26} has emphasized the use of elementbased mass fractions which become conserved scalars because chemistry does not destroy atoms but only changes molecules. This allows us to consider general chemical kinetics without the use of the onestep assumption. Define the element mass fraction for the atom identified by integer k as $ Y k \u2261 \Sigma m = 1 N a m , k Y m W k / W m $, where a_{m,k}, W_{k}, and W_{m} are the integer number of k atoms in molecule m, the atomic weight of k, and the molecular weight of m, respectively. Then, Y_{k} is a conserved scalar satisfying the homogeneous forms of the differential equations given by Eqs. (5) and (35) for the unsteady and steady states, respectively. Defining Z ≡ (Y_{k} − Y_{k,−∞})/(Y_{k,∞} − Y_{k,−∞}), it satisfies these equations with Z varying from 0 to 1. For any k, the steadystate solution is again Z = J(η)/J(∞).
In the steady state, the reaction rate in Eq. (38) is determined from the mass fractions using the known linear relations among temperature (from enthalpy for constant c_{p}), mass fractions, and the conserved scalars Z, α, and β. So, a solution can be found in Zspace. For the steady state with fast chemical kinetics, $ \omega \u0307 m ( Z ) $ will have a significant value within a narrow region in Zspace around the stoichiometric value. On both sides of that narrow region, Y_{m} will be linear in Z.
Figure 9(a) shows that the burning region with the local density minimum experiences larger changes in y for a given change in η. Subsequently, we see in the streamline projections of Figs. 9(b) and 9(c) that streamlines in the negativey upward flow region are stretched vertically more than in the downward flow. The streamline projections are the same for any x, y plane; that is, they do not depend on the z value. Likewise, the streamline projection onto any the y, z plane is identical and without x dependence. However, the streamline projection onto an x, z plane will vary depending on the value of y (or equivalently η). The effects of different ambient temperatures and density for the two opposing streams, obviously, will have an impact on the variations of streamline projections onto the x–z plane for both reacting and nonreacting counterflows. Also, it will affect wallstagnation flows where wall temperature and ambient temperature differ, causing density gradients.
V. 3D WALL STAGNATIONPOINT FLOW
The solutions for $v$ and p remain in the same form as given by Eqs. (8) and (27). While the forms for counterflow and wall stagnation flow are the same or similar, the changes in boundary conditions for f_{1} and f_{2} due to the noslip wall will affect the solutions. $ h w $ is the gas enthalpy at the wall.
If ρ = 1, μ = 1, and η = y, we may obtain the solutions for the incompressible case. Howarth^{15} has provided the incompressible steady solutions for the continuity and momentum equations but not for the energy equation.
Figures 10–12 give the computational wallstagnationflow results for various values of S_{1}, S_{2}, Pr, and $ \rho w $ = h_{∞}/$ h w $. Again, the rate of strain has modest affect on the scalar fields and $v$. Significant influence is seen on the u and $w$ velocity components which each increases with normal strain values for their respective direction. The term $ S 1,2 [ h \u0303 \u2212 ( f 1,2 \u2032 ) 2 ] $ in Eq. (31) can be large, especially near the wall. Still again, the results are readily extended since the values for u/x, and $w$/z can be interchanged with the values for $w$/z and u/x when S_{1} and S_{2} are replaced by 1 − S_{1} and 1 − S_{2}, respectively. For the case of a hot wall, Fig. 12 shows that all of the velocity components u, $v$, and $w$ have overshoots in the lowdensity region near that hot wall.
In similar fashion to the variabledensity counterflow case, the extension of the results to a case with inflow from both the y and x (or z) directions would not describe a plausible flow configuration.
VI. CONCLUSIONS
Viscous counterflows and wall stagnation flows are analyzed with threedimensional normal strain rates. Reacting (i.e., with diffusion flames) and nonreacting counterflows are examined. Stagnation flows with hot and cold walls are also studied. A similar system of the NavierStokes equations coupled with equations for scalar transport is developed, and exact solutions are obtained both inside and outside the viscous layer; i.e., the boundarylayer approximation is not required. The second derivatives of velocity go to zero outside of the viscous layer although first derivatives remain and asymptote to constants. Consequently, viscous force (per volume) asymptotes to zero and the NavierStokes equations asymptote to the Euler equations. Variable density, temperature, and composition are considered. Results for planar flows and axisymmetric flows are obtained as limits here. Velocity components u, $v$, and $w$ are each odd functions of x, y, and z, respectively. The scalar functions are even functions of both x and z. Terms of O(x^{2}, xz, z^{2}) are neglected. $v$(y) is determined for the infinite range of y.
For the steady and unsteady incompressible counterflows, analytical solutions are obtained for the flow field and the scalar fields subject to heat and mass transfer. In steady, variabledensity configurations (both reacting and nonreacting), a set of ODEs govern the two transverse velocity profiles. Each of the three velocity components as well as the diffusion rates for mass, momentum, and energy depends on the two normal strain rates parallel to the counterflow interface or the wall and thereby not merely on the sum of those two strain rates. The effect of strain distribution is generally greater on the transverse velocity components, u and $w$, than on the incoming velocity component, $v$, and the scalar variables. The incompressible counterflow is the only case where the diffusion rate and the velocity component in the counterflow direction are not affected by the distribution of the strain rate between the two transverse directions and depend only on the sum of those two strain rates. The velocity profiles and the scalar profiles are shown to depend on ambient temperature (or equivalently density) values and the Prandtl number as well as the strain rates.
The results obtained here for both stagnationpoint flows and counterflows agree exactly in the limits of twodimensional flows (planar and axisymmetric) with the well known literature.^{1,6,13,14,17–19} It also agrees with Howarth’s results for threedimensional, incompressible, steady wallstagnation flow.^{15}
The cited twodimensional solutions and this new threedimensional solution are exact solutions to the NavierStokes equations for a region neighboring the stagnation point. For a flat interface of the opposing streams in counterflow or a flat wall in wallstagnation flow, that neighborhood becomes infinite in size. Essentially, we have an acceptable solution if the ratio of the neighborhood size to the interface radius is much smaller than unity. Of course, improvements in the presentation here can be made by better representations of transport and physical property values, chemical kinetic descriptions, and in some cases equations of state.
For thin diffusion flames, the location, burning rate, and peak temperature are readily obtained in the infinitekinetics limit. Important corrections are shown of the existing literature which is based on a constantdensity assumption. For counterflows with flames and stagnation layers with hot walls, velocity overshoots are seen in the viscous layer for all three velocity components. The overshoot of velocity $v$ in the incoming direction is driven by hotgas expansion through the continuity equation. Then, for the turning flow, overshoots in the u and $w$ velocity components also result. These overshoots also occur in the axisymmetric and planar limits although they were not recognized in prior studies on gaseous combustion.^{1,6,16–19} The overshoot in the $v$ velocity was presented for spray flames,^{20} and, to a small extent, velocity overshoot appeared for combustion at supercritical pressure,^{22} however without much discussion and no examination of the transverse velocity components. Here, solutions are found for a full range of the distribution of normal strain rates between the two transverse directions, various Prandtl number values, and various ambient (or wall) temperatures. The velocity overshoots for all three components show that large velocity gradients and vorticity of opposite directions are developing. These occurrences can have significant consequence for hydrodynamic stability and the development of turbulence. Also, it can add to the effect of flame stretch and subsequent extinction.
In steady counterflow and wallstagnation flow, streamline projections on the x, y plane are the same for any z value. Similarly, projections on the y, z plane are the same for any x value. However, with variable density, projections onto an x, z plane will vary with y. A local reduction in density results in a local increase in the local strainrate ratio beyond the ambient strainrate ratio, thereby turning the flow vector locally even more in the direction of the greater strain rate.
Although viscous force goes to zero asymptotically with increasing distance from the wall or interface layer, the viscous dissipation rate asymptotes to a constant value.
ACKNOWLEDGMENTS
This research was supported by the Air Force Office of Scientific Research under Grant No. FA95501810392 with Dr. Mitat Birkan as the scientific officer. Information about existing literature from Professors D. Papamoschou, A. L. Sanchez, and F. A. Williams has been very helpful. The guidance from Professors T. Georgiou and A. Sideris on optimal use of Matlab for numerical integration is greatly appreciated.