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Turbulent Non-Premixed Combustion
Combustion Summer School
Prof. Dr.-Ing. Heinz Pitsch
2018
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Course Overview
2
• Turbulence
• Turbulent Premixed Combustion
• Turbulent Non-Premixed Combustion
• Turbulent Combustion Modeling
• Applications
• Laminar Jet Diffusion Flames • Turbulent Jet Diffusion Flames
Part II: Turbulent Combustion
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flame length L
fuel air
Laminar Jet Diffusion Flames
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Laminar Jet Diffusion Flame
• Fuel enters into the combustion chamber as a round jet
• Forming mixture is ignited • Example: Flame of a gas lighter
− Only stable if dimensions are small − Dimensions too large: flickering due to
influence of gravity − Increasing the jet momentum → Reduction
of the relative importance of gravity (buoyancy) in favor of momentum forces
− At high velocities, hydrodynamic instabilities gain increasing importance: laminar-turbulent transition
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Laminar Diffusion Fame: Influence of Gravity
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1g 0g
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Laminar Jet Diffusion Flame (Governing Equations)
• Starting point: Conservation equations for stationary axisymmetric boundary layer flow without buoyancy
• Continuity:
• Momentum equation in z-direction
• Mixture fraction
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flame length L
fuel air
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Laminar Jet Diffusion Flame (Assumptions + BC)
• Schmidt number Sc = μ/ρD • Farfield area
− r → ∞: uz = ur = 0 − From z-momentum equation dp/dz = 0
• Boundary layer flow:
• Incompressible round jet − Quiescent ambient − Constant density − No buoyancy → Similarity solution
• Simularity coordinate η = r/z (Schlichting, „Boundary Layer Theory“)
7
0 flame length L
fuel air
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Laminar Jet Diffusion Flame (Similarity Coordinates)
• If density not constant → Transformation
• a: Distance of the virtual origin of the jet from the nozzle exit
• For ρ = const. und a → 0
• Implies linear spreading of the roung jet
8
flame length L
fuel air
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Laminar Jet Diffusion Flame (Stream Function)
• Introduction of a stream function
→ Continuity equation identically satisfied
• Applying the transformation rules to the convective terms in the momentum and mixture fraction equations yields
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Laminar Jet Diffusion Flame (Transformation Rules)
• Applying the transformation rules to the convective terms in the momentum and mixture fraction equations yields
• The diffusive terms become
• C: Chapman-Rubesin-Parameter
• For constant density (with η = r/ζ and μ = μ∞): C = 1 10
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Laminar Jet Diffusion Flame (Non-dim. Stream Func.)
• Formal transformation of the momentum and concentration equations and assumption that C = f(ζ,η)
• With ansatz for non-dimensional stream function F for the velocities follows
• uz und ur can be expressed as a function of the nondimensional stream function F and its derivatives
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Laminar Jet Diffusion Flame (Transformation)
• From the momentum equation
• Similarity solution only exists, if F ≠ f(ζ) • Then, uz is proportional to 1/ζ (see previous slide)
→ velocity decreases linearly with 1/(z + a) • Prerequesites: Boundary conditions and C are independent of z
(e. g. uz = 0 and ur = 0 for η → 0)
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Laminar Jet Diffusion Flame (Resulting Equations)
• Equation for the nondimensional stream function
• Let ω = Z(z,r)/Za(z), ratio of the mixture fraction Za(z) to its value at r = 0 • Applying the same transformations to the ω-equation yields
• In case of a similarity solution
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Laminar Jet Diffusion Flame (Analytic Solution)
• Integration for C = const. yields:
where γ is integration constant
• The assumption C = const. Holds if and ρμ/ρm μ∞ = const.
• C = const. Often not a good assumption, since
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Laminar Jet Diffusion Flame (Integration Constant γ)
• Constant of integration γ can be determined from the condition that the jet momentum is independent of ζ
• Substitution of the solution into the momentum balance yields
• ρ0: density of the fuel stream • Reynolds number Re = uz,0d/ν∞
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Laminar Jet Diffusion Flame (Centerline Mixt. Fraction)
• Analogously for the mixture fraction (with Z0 = 1)
→ Mixture fraction on the centerline Za(z) = Z(z,r=0):
→ Za decreases with 1/ζ (as the velocity)
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flame length L
fuel air
Laminar Jet Diffusion Flame (Flame Length)
• Determination of the flame contour r as function of z from the condition
• Flame contour intersects centerline, r = 0, if Za = Zst
• Corresponding value of z defines the flame length
• Valid for laminar jet flames without buoyancy
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Laminar Jet Diffusion Flame
• For a given nozzle diameter, L increases linearly with the Reynolds number Re
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flame length L
fuel air
Reynolds number Re
transition fully developed turbulent flame
laminar flame
flam
e le
ngth
L/d
Sct=0,72
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Course Overview
19
• Turbulence
• Turbulent Premixed Combustion
• Turbulent Non-Premixed Combustion
• Turbulent Combustion Modeling
• Applications
• Laminar Jet Diffusion Flames • Turbulent Jet Diffusion Flames
Part II: Turbulent Combustion
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Turbulent Jet Diffusion Flame
• Shear flow at nozzle exit • Flow instabilities
(Kelvin-Helmholtz-instabilities) → laminar-turbulent transition
• Ring shaped turbulent shear layer propagates in radial direction
• Merging after 10 to 15 nozzle diameters downstream
• Streamlines are parallel in potential core • Velocity profile reaches self similar
state after 20-30 nozzle diameters
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Round Turbulent Diffusion Flame
• Linear reduction of velocity along central axis • Linear increase of jet width • Assumption: fast chemical reaction
→ Scalar quantities such as temperature, concentration and density as function of mixture fraction Z
• Turbulent flow with variable density → Favre-averaged boundary layer equations
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Linear Propagation of (turbulent) Jet
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Round Turbulent Diffusion Flame
• Assumptions: − Axisymmetric jet flame − Neglecting buoyancy − Neglecting molecular transport as compared
to turbulent transport − Turbulent transport modeled by
Gradient Transport model − Sct = νt/Dt
• Using Favre averaging and the the boundary layer assumption we obtain a system of two-dimensional axisymmetric equations
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Round Turbulent Diffusion Flame
• Continuity equation
• Momentum equation in z-direction
• Mean mixture fraction
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Round Turbulent Diffusion Flame
• Requires solving of equations for k and ε to determine νt • Round turbulent jet: νt approximately constant • Analogous for round laminar jet:
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Turbulent Laminar
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Round Turbulent Diffusion Flame
• Special case: Jet in quiescent ambient − Treatment of turbulent equations like those in a laminar round jet case − Using the laminar theory
• Similarity coordinate
• Chapman-Rubesin-Parameter
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Turbulent Laminar
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Round Turbulent Diffusion Flame
• Turbulent Chapman-Rubesin-Parameter approximately constant→
• Integration constant γ, containing fuel density and reference viscosity
• The Favre-averaged velocity decreases proportional to 1/ζ = 1/(z + a), just like in the laminar case
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Round Turbulent Diffusion Flame
• Mean mixture fraction with
→ Mixture fraction decreases proportional to 1/(z + a) on the jet axis
→ Progression of profiles along jet axis resembles those of the laminar case • Also applies to the contour of the stoichiometric mixture
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Round Turbulent Diffusion Flame
• Flame length L of round turbulent diffusion flame: Distance z from the nozzle, where the mean mixture fraction on the axis equals Zst
• Comparison with experimental correlations (Hawthorne, Weddel and Hottel (1949)) − With uz,0d/νt,ref = 70 and Sct=0,72 − Complete agreement for C = (ρ0 ρst)1/2/ρ∞
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Round Turbulent Diffusion Flame
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const.
≈ 70
linear
Reynolds number Re
transition fully developed turbulent flame
laminar flame
flam
e le
ngth
L/d
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Experimental Data: Round Turbulent Diffusion Flame
• Comparison of experimental results and simulations with chemical equilibrium
• Concentration of radicals and emissions cannot be described by infinitely fast chemistry
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Summary
32
• Turbulence
• Turbulent Premixed Combustion
• Turbulent Non-Premixed Combustion
• Turbulent Combustion Modeling
• Applications
• Laminar Jet Diffusion Flames • Turbulent Jet Diffusion Flames
Part II: Turbulent Combustion