The fractional rate of change of a reactant can be written as

For most hydrocarbon flame or reacting systems the overall order of reaction is about 2, E/R is approximately 20,000 K, and the flame temperature is about 2000 K. Thus.

In this hypothetical simplified problem, one assumes further that the temperature T fluctuates with time around some mean represented by the form

Dividing the two expressions, one obtains

Obviously, then, for small fluctuations

The expression for the mean rate is written as

But recall,

Examining the third term, it is apparent that

If the amplitude of the temperature fluctuations is of the order of 10% of the mean temperature, one can take a_n ≈ 0.1; and if the fluctuations are considered sinusoidal, then

Thus for the example being discussed,

This result could be improved by assuming a more appropriate distribution function of T′ instead of a simple sinusoidal fluctuation; however, this example—even with its assumptions—usefully illustrates the problem. Normally, probability distribution functions are chosen. If the concentrations and temperatures are correlated, the rate expression becomes complicated. Bilger [50] presented a form of a two-component mean-reaction rate when it is expanded about the mean states, as follows:

Most flows have at least one characteristic velocity, U, and one characteristic length scale, L, of the device in which the flow takes place. In addition there is at least one representative density ρ₀ and one characteristic temperature T₀, usually the unburned condition when considering combustion phenomena. Thus, a characteristic kinematic viscosity ν₀ ≡ μ₀/ρ₀ can be defined, where μ₀ is the coefficient of viscosity at the characteristic temperature T₀. The Reynolds number for the system is then Re = UL/ν₀. It is interesting that ν is approximately proportional to T². Thus, an increase in temperature by a factor of three or more, modest by combustion standards, means a drop in Re by an order of magnitude. Thus, energy release can dampen turbulent fluctuations. The kinematic viscosity ν is inversely proportional to the pressure P, and changes in P are usually small; the effects of such changes in ν typically are much less than those due to changes in T.

Even though the Reynolds number gives some measure of turbulent phenomena, flow quantities characteristic of turbulence itself are of more direct relevance to modeling turbulent reacting systems. The turbulent kinetic energy q¯ may be assigned a representative value q¯0 at a suitable reference point. The relative intensity of the turbulence is then characterized by either q¯0/(12U2) or U′/U, where U′=(2q¯0)1/2 is a representative root-mean-square velocity fluctuation. Weak turbulence corresponds to U′/U<1 and intense turbulence has U′/U of the order unity.

Although a continuous distribution of length scales is associated with the turbulent fluctuations of velocity components and of state variables (P, ρ, T), it is useful to focus on two widely disparate lengths that determine separate effects in turbulent flows. First, there is a length l₀, which characterizes the large eddies, those of low frequencies and long wavelengths; this length is sometimes referred to as the integral scale. Experimentally, l₀ can be defined as a length beyond which various fluid-mechanical quantities become essentially uncorrelated; typically, l₀ is less than L but of the same order of magnitude. This length can be used in conjunction with U′ to define a turbulent Reynolds number

The second length scale characterizing turbulence is that over which molecular effects are significant; it can be introduced in terms of a representative rate of dissipation of velocity fluctuations, essentially the rate of dissipation of the turbulent kinetic energy. This rate of dissipation, which is given by the symbol ε₀, is

This rate estimate corresponds to the idea that the time scale over which velocity fluctuations (turbulent kinetic energy) decay by a factor of (1/e) is the order of the turning time of a large eddy. The rate ε₀ increases with turbulent kinetic energy (which is due principally to the large-scale turbulence) and decreases with increasing size of the large-scale eddies. For the small scales at which molecular dissipation occurs, the relevant parameters are the kinematic viscosity, which causes the dissipation, and the rate of dissipation. The only length scale that can be constructed from these two parameters is the so-called Kolmogorov length (scale):

However, note that

Therefore

This length is representative of the dimension at which dissipation occurs and defines a cut-off of the turbulence spectrum. For large R_l there is a large spread of the two extreme lengths characterizing turbulence. This spread is reduced with the increasing temperature found in combustion from the consequent increase in ν₀.

Considerations analogous to those for velocity apply to scalar fields as well, and lengths analogous to l_k have been introduced for these fields. They differ from l_k by factors involving the Prandtl and Schmidt numbers, which differ relatively little from unity for representative gas mixtures. Therefore, to a first approximation for gases, l_k may be used for all fields, and there is no need to introduce any new corresponding lengths.

An additional length, intermediate in size between l₀ and l_k, which often arises in formulations of equations for average quantities in turbulent flows is the Taylor length (λ), which is representative of the dimension over which strain occurs in a particular viscous medium. The strain can be written as (U′/l₀). As before, the length that can be constructed between the strain and the viscous forces is

In a sense, the Taylor microscale is similar to an average of the other scales, l₀ and l_k, but heavily weighted toward l_k.

Recall that there are length scales associated with laminar flame structures in reacting flows. One is the characteristic thickness of a premixed flame, δ_L, given by

The derivation is, of course, consistent with the characteristic velocity in the flame speed problem. This velocity is obviously the laminar flame speed itself, so that

As discussed in an earlier section, δ_L is the characteristic length of the flame and includes the thermal preheat region and that associated with the zone of rapid chemical reaction. This reaction zone is the rapid heat release flame segment at the high-temperature end of the flame. The earlier discussion of flame structure from detailed chemical kinetic mechanisms revealed that the heat release zone need not be narrow compared with the preheat zone. Nevertheless, the magnitude of δ_L does not change, no matter what the analysis of the flame structure is. It is then possible to specify the characteristic time of the chemical reaction in this context to be

It may be expected, then, that the nature of the various turbulent flows, and indeed the structures of turbulent flames, may differ considerably and their characterization would depend on the comparison of these chemical and flow scales in a manner specified by the following inequalities and designated flame type:

The nature, or more precisely the structure, of a particular turbulent flame implied by these inequalities cannot be exactly established at this time. The reason is that values of δ_L, l_k, λ, or l₀ cannot be explicitly measured under a given flow condition or analytically estimated. Many of the early experiments with turbulent flames appear to have operated under the condition δ_L < l_k, so the early theories that developed specified this condition in expressions for S_T. The flow conditions under which δ_L would indeed be less than l_k have been explored analytically in detail and will be discussed subsequently.

To expand on the understanding of the physical nature of turbulent flames, it is also beneficial to look closely at the problem from a chemical point of view, exploring how heat release and its rate affect turbulent flame structure.

One begins with the characteristic time for chemical reaction designated τ_c, which was defined earlier. (Note that this time would be appropriate whether a flame existed or not.) Generally, in considering turbulent reacting flows, chemical lengths are constructed to be Uτ_c or U′τ_c. Then comparison of an appropriate chemical length with a fluid dynamic length provides a nondimensional parameter that has a bearing on the relative rate of reaction. Nondimensional numbers of this type are called Damkohler numbers and are conventionally given the symbol Da. An example appropriate to the considerations here is

For large Damkohler numbers, the chemistry is fast (i.e., reaction time is short) and reaction sheets of various wrinkled types may occur. For small Da numbers, the chemistry is slow and well-stirred flames may occur.

Two other nondimensional numbers relevant to the chemical reaction aspect of this problem [51] were introduced by Frank–Kamenetskii and others. These Frank–Kamenetskii numbers (FK) are the nondimensional heat release FK₁ ≡ (Q_p/c_pT_f), where Q_p is the chemical heat release of the mixture and T_f is the flame (or reaction) temperature; and the nondimensional activation energy FK₂ ≡ (T_a/T_f), where the activation temperature T_a = (E_A/R). Combustion, in general, and turbulent combustion, in particular, are typically characterized by large values of these numbers. When FK₁ is large, chemistry is likely to have a large influence on turbulence. When FK₂ is large, the rate of reaction depends strongly on the temperature. It is usually true that the larger the FK₂, the thinner will be the region in which the principal chemistry occurs. Thus, irrespective of the value of the Damkohler number, reaction zones tend to be found in thin, convoluted sheets in turbulent flows, for both premixed and non-premixed systems having large FK₂. For premixed flames, the thickness of the reaction region has been shown to be of the order δ_L/FK₂. Different relative sizes of δ_L/FK₂ and fluid-mechanical lengths, therefore, may introduce additional classes of turbulent reacting flows.

The flames themselves can alter the turbulence. In simple open Bunsen flames whose tube Reynolds number indicates that the flow is in the turbulent regime, some results have shown that the temperature effects on the viscosity are such that the resulting flame structure is completely laminar. Similarly, for a completely laminar flow in which a simple wire is oscillated near the flame surface, a wrinkled flame can be obtained (Figure 4.41). Certainly, this example is relevant to δ_L < l_k; that is, a wrinkled flame. Nevertheless, most open flames created by a turbulent fuel jet exhibit a wrinkled flame type of structure. Indeed, short-duration Schlieren photographs suggest that these flames have continuous surfaces. Measurements of flames such as that shown in Figures. 4.42(a) and (b) have been taken at different time intervals, and the instantaneous flame shapes verify the continuous wrinkled flame structure. A plot of these instantaneous surface measurements results in a thick flame region (Figure 4.43), just as the eye would visualize that a larger number of these measurements would result in a thick flame. Indeed, turbulent premixed flames are described as bushy flames. The thickness of this turbulent flame zone appears to be related to the scale of turbulence. Essentially, this case becomes that of severe wrinkling and is categorized by l_k < δ_L < λ. Increased turbulence changes the character of the flame wrinkling, and flamelets begin to form. These flame elements take on the character of a fluid-mechanical vortex rather than a simple distorted wrinkled front, and this case is specified by λ < δ_L < l₀. For δ_L << l₀, some of the flamelets fragment from the front, and the flame zone becomes highly wrinkled with pockets of combustion. To this point, the flame is considered practically contiguous. When l₀ < δ_L, contiguous flames no longer exist, and a distributed reaction front forms. Under these conditions, the fluid mixing processes are rapid with respect to the chemical reaction time, and the reaction zone essentially approaches the condition of a stirred reactor. In such a reaction zone, products and reactants are continuously intermixed.

For a better understanding of this type of flame occurrence and for more explicit conditions that define each of these turbulent flame types, it is necessary to introduce the flame stretch concept. This will be done shortly, at which time the regions will be more clearly defined with respect to chemical and flow rates with a graph that relates the nondimensional turbulent intensity, Reynolds numbers, Damkohler number, and characteristic lengths l.

First, however, consider that in turbulent Bunsen flames the axial component of the mean velocity along the centerline remains almost constant with height above the burner; but away from the centerline, the axial mean velocity increases with height. The radial outflow component increases with distance from the centerline and reaches a peak outside the flame. Both axial and radial components of turbulent velocity fluctuations show a complex variation with position and include peaks and troughs in the flame zone. Thus, there are indications of both generation and removal of turbulence within the flame. With increasing height above the burner, the Reynolds shear stress decays from that corresponding to an initial pipe flow profile.

In all flames there is a large increase in velocity as the gases enter the burned gas state. Thus, it should not be surprising that the heat release itself can play a role in inducing turbulence. Such velocity changes in a fixed combustion configuration can cause shear effects that contribute to the turbulence phenomenon. There is no better example of some of these aspects than the case in which turbulent flames are stabilized in ducted systems. The mean axial velocity field of ducted flames involves considerable acceleration resulting from gas expansion engendered by heat release. Typically, the axial velocity of the unburned gas doubles before it is entrained into the flame, and the velocity at the centerline at least doubles again. Large mean velocity gradients are therefore produced. The streamlines in the unburned gas are deflected away from the flame.

The growth of axial turbulence in the flame zone of these ducted systems is attributed to the mean velocity gradient resulting from the combustion. The production of turbulence energy by shear depends on the product of the mean velocity gradient and the Reynolds stress. Such stresses provide the most plausible mechanism for the modest growth in turbulence observed.

Now it is important to stress that whereas the laminar flame speed is a unique thermochemical property of a fuel–oxidizer mixture ratio, a turbulent flame speed is a function not only of the fuel–oxidizer mixture ratio but also of the flow characteristics and experimental configuration. Thus, one encounters great difficulty in correlating the experimental data of various investigators. In a sense, there is no flame speed in a turbulent stream. Essentially, as a flow field is made turbulent for a given experimental configuration, the mass consumption rate (and hence the rate of energy release) of the fuel–oxidizer mixture increases. Therefore, some researchers have found it convenient to define a turbulent flame speed S_T as the mean mass flux per unit area (in a coordinate system fixed to the time-averaged motion of the flame) divided by the unburned gas density ρ₀. The area chosen is the smoothed surface of the time-averaged flame zone. However, this zone is thick and curved; thus, the choice of an area near the unburned gas edge can give a different result from one in which a flame position is taken in the center or the burned-gas side of the bushy flame. Therefore, a great deal of uncertainty is associated with the various experimental values of S_T reported. Nevertheless, definite trends have been reported. These trends can be summarized as follows:

The definition of the flame speed as the mass flux through the flame per unit area of the flame divided by the unburned gas density ρ₀ is useful for turbulent nonstationary and oblique flames as well.

Now with regard to stretch, consider first a plane oblique flame. Because of the increase in velocity demanded by continuity, a streamline through such an oblique flame is deflected toward the direction of the normal to the flame surface. The velocity vector may be broken up into a component normal to the flame wave and a component tangential to the wave (Figure 4.44). Because of the energy release, the continuity of mass requires that the normal component increase on the burned gas side while, of course, the tangential component remains the same. A consequence of the tangential velocity is that fluid elements in the oblique flame surface move along this surface. If the surface is curved, adjacent points traveling along the flame surface may move either farther apart (flame stretch) or closer together (flame compression).

An oblique flame is curved if the velocity U of the approach flow varies in a direction y perpendicular to the direction of the approach flow. Strehlow showed that the quantity

Since the concern here is with the destruction of a contiguous laminar flame in a turbulent field, consideration must also be given to certain inherent instabilities in laminar flames themselves. There is a fundamental hydrodynamic instability as well as an instability arising from the fact that mass and heat can diffuse at different rates; that is, the Lewis number (Le) is nonunity. In the latter mechanism, a flame instability can occur when the Le number (α/D) is less than 1.

Consider initially the hydrodynamic instability—that is, the one due to the flow—first described by Darrieus [55], Landau [56], and Markstein [57]. If no wrinkle occurs in a laminar flame, the flame speed S_L is equal to the upstream unburned gas velocity U₀. But if a minor wrinkle occurs in a laminar flame, the approach flow streamlines will either diverge or converge, as shown in Figure 4.45. Considering the two middle streamlines, one notes that because of the curvature owing to the wrinkle, the normal component of the velocity with respect to the flame is less than U₀. Thus, the streamlines diverge as they enter the wrinkled flame front. Since there must be continuity of mass between the streamlines, the unburned gas velocity at the front must decrease owing to the increase of area. Since S_L is now greater than the velocity of unburned approaching gas, the flame moves farther downstream and the wrinkle is accentuated. For similar reasons, between another pair of streamlines if the unburned gas velocity increases near the flame front, the flame bows in the upstream direction. It is not clear why these instabilities do not keep growing. Some have attributed the growth limit to nonlinear effects that arise in hydrodynamics.

When the Lewis number is nonunity, the mass diffusivity can be greater than the thermal diffusivity. This discrepancy in diffusivities is important with respect to the reactant that limits the reaction. Ignoring the hydrodynamic instability, consider again the condition between a pair of streamlines entering a wrinkle in a laminar flame. This time, however, look more closely at the flame structure that these streamlines encompass, noting that the limiting reactant will diffuse into the flame zone faster than heat can diffuse from the flame zone into the unburned mixture. Thus, the flame temperature rises, the flame speed increases, and the flame wrinkles bow further in the downstream direction. The result is a flame that looks much like the flame depicted for the hydrodynamic instability in Figure 4.45. The flame surface breaks up continuously into new cells in a chaotic manner, as photographed by Markstein [54]. There appears to be, however, a higher-order stabilizing effect. The fact that the phenomenon is controlled by a limiting reactant means that this cellular condition can occur when the unburned premixed gas mixture is either fuel-rich or fuel-lean. It should not be surprising, then, that the most susceptible mixture would be a lean hydrogen–air system.

Earlier it was stated that the structure of a turbulent velocity field may be presented in terms of two parameters—the scale and the intensity of turbulence. The intensity was defined as the square root of the turbulent kinetic energy, which essentially gives a root-mean-square velocity fluctuation U′. Three length scales were defined: the integral scale l₀, which characterizes the large eddies; the Taylor microscale λ, which is obtained from the rate of strain; and the Kolmogorov microscale l_k, which typifies the smallest dissipative eddies. These length scales and the intensity can be combined to form not one, but three turbulent Reynolds numbers: R_l = U′l₀/ν, R_λ = U′λ/ν, and R_k = U′l_k/ν. From the relationship between l₀, λ, and l_k previously derived it is found that Rl≈Rλ2≈Rk4.

There is now sufficient information to relate the Damkohler number Da and the length ratios l₀/δ_L, l_k/δ_L and l₀/l_k to a nondimensional velocity ratio U′/S_L and the three turbulence Reynolds numbers. The complex relationships are given in Figure 4.46 and are informative. The right-hand side of the figure has R_λ > 100 and ensures the length-scale separation that is characteristic of high Reynolds number behavior. The largest Damkohler numbers are found in the bottom right corner of the figure.

Using this graph and the relationship it contains, one can now address the question of whether and under what conditions a laminar flame can exist in a turbulent flow. As before, if allowance is made for flame front curvature effects, a laminar flame can be considered stable to a disturbance of sufficiently short wavelength; however, intense shear can lead to extinction. From solutions of the laminar flame equations in an imposed shear flow, Klimov [53] and Williams [54] showed that a conventional propagating flame may exist only if the stretch factor K₂ is less than a critical value of unity. Modeling the area change term in the stretch expression as

But as shown earlier

Thus, the criterion to be satisfied if a laminar flame is to exist in a turbulent flow is that the laminar flame thickness δ_L be less than the Kolmogorov microscale l_k of the turbulence.

The heavy line in Figure 4.46 indicates the conditions δ_L = l_k. This line is drawn in this fashion since

Thus, for (U′/S_L) = 1, R_λ = 1; and for (U′/S_L) = 10, R_λ = 100. The other Reynolds numbers follow from Rk4=Rλ2=Rl.

Below and to the right of this line, the Klimov–Williams criterion is satisfied and wrinkled laminar flames may occur. The figure shows that this region includes both large and small values of turbulence Reynolds numbers and velocity ratios (U′/S_L) both greater and less than 1, but predominantly large Da.

Above and to the left of the criterion line is the region in which l_k < δ_L. According to the Klimov–Williams criterion, the turbulent velocity gradients in this region, or perhaps in a region defined with respect to any of the characteristic lengths, are sufficiently intense that they may destroy a laminar flame. The figure shows U′ ≥ S_L in this region and Da is predominantly small. At the highest Reynolds numbers the region is entered only for intense turbulence, U′ > S_L. The region has been considered a distributed reaction zone in which reactants and products are somewhat uniformly dispersed throughout the flame front. Reactions are still fast everywhere, so that unburned mixture near the burned-gas side of the flame is completely burned before it leaves what would be considered the flame front. An instantaneous temperature measurement in this flame would yield a normal probability density function—more important, one that is not bimodal.

In his attempts to analyze the early experimental data, Damkohler [58] considered that large-scale, low-intensity turbulence simply distorts the laminar flame while the transport properties remain the same; thus, the laminar flame structure would not be affected. Essentially, his concept covered the range of the wrinkled and severely wrinkled flame cases defined earlier. Whereas a planar laminar flame would appear as a simple Bunsen cone, that cone is distorted by turbulence as shown in Figure 4.43. It is apparent, then, that the area of the laminar flame will increase owing to a turbulent field. Thus, Damkohler [58] proposed for large-scale, small-intensity turbulence that

Many groups of experimental data have been evaluated by semiempirical correlations of the type

The first expression here is similar to the Damkohler result for A and B equal to 1. Since the turbulent exchange coefficient (eddy diffusivity) correlates well with l₀U′ for tube flow, and indeed, l₀ is essentially constant for the tube flow characteristically used for turbulent premixed flame studies, it follows that

Schelkin [59] also considered large-scale, small-intensity turbulence. He assumed that flame surfaces distort into cones with bases proportional to the square of the average eddy diameter (i.e., proportional to l₀). The height of the cone was assumed proportional to U′ and to the time t during which an element of the wave is associated with an eddy. Thus, time can be taken as equal to (l₀/S_L). Schelkin then proposed that the ratio of S_T/S_L (average) equals the ratio of the average cone area to the cone base. From the geometry thus defined,

For large values of (U′/SL), that is, high-intensity turbulence, the preceding expression reduces to that developed by Damkohler: ST∼U′.

A more rigorous development of wrinkled turbulent flames led Clavin and Williams [60] to the following result where isotropic turbulence is assumed:

This result differs from Schelkin's heuristic approach only by the factor of two in the second term. The Clavin-Williams expression is essentially restricted to the case of (U′/SL)<<1. For small (U′/SL), the Clavin-Williams expression simplifies to

Using a direct numerical simulation, Yakhot [62] proposed the relation

For small-scale, high-intensity turbulence, Damkohler reasoned that the transport properties of the flame are altered from laminar kinetic theory viscosity ν₀ to the turbulent exchange coefficient ε so that

This expression derives from S_L ∼ α^1/2 ∼ ν^1/2. Then, realizing that ε∼U′l0,

Schelkin [59] also extended Damkohler's model by starting from the fact that the transport in a turbulent flame could be made up of molecular movements (laminar λ_L) and turbulent movements, so that

The Damkohler turbulent exchange coefficient ε is the same as ν_T, so that both expressions are similar, particularly in that for high-intensity turbulence ε >> ν. The Damkohler result for small-scale, high-intensity turbulence that

The general data representation in Figure 4.47 shows a rapid rise of (S_T/S_L) for values of (U′/SL)<<1. It is apparent from the discussion to this point that as (U′/SL) becomes greater than 1, the character of the turbulent flame varies; and under the appropriate turbulent variables, it can change as depicted in Figure 4.48, which essentially comes from Borghi [63] as presented by Abdel-Gayed et al. [64].