Fanno flow
Fanno flow, named after the Italian engineer Gino Fanno, is a model in fluid dynamics describing the steady, one-dimensional, adiabatic flow of a compressible ideal gas through a constant-area duct with wall friction but no heat transfer.[1] This flow represents frictional effects in confined channels, where wall shear stress influences thermodynamic and flow properties, often leading to choking when the Mach number reaches unity at the duct exit.[2] The fundamental assumptions of Fanno flow include steady-state conditions, negligible heat conduction through the walls, fully developed velocity profiles, and constant specific heats for the ideal gas.[1] Governing equations derive from conservation of mass, momentum, and energy: mass continuity ensures constant mass flux (\rho V A = \constant), the energy equation maintains constant stagnation temperature (T_0 = \constant), and the momentum equation incorporates frictional losses via the Darcy friction factor f, relating changes in Mach number M to duct length through \frac{f L}{D_h} = \frac{1 - M^2}{\gamma M^2} + \frac{\gamma + 1}{2\gamma} \ln \left( \frac{(\gamma + 1) M^2}{2(1 + \frac{\gamma - 1}{2} M^2)} \right), where \gamma is the specific heat ratio and D_h is the hydraulic diameter.[3] These relations allow prediction of variations in pressure, temperature, and velocity along the duct. In subsonic Fanno flow (inlet M < 1), friction accelerates the flow, decreasing static pressure and temperature while increasing Mach number toward 1, potentially causing choking if the duct is sufficiently long.[3] Conversely, supersonic inlet flow (M > 1) decelerates due to friction, with pressure and temperature rising as Mach number decreases to 1, also risking choking at a maximum duct length.[3] Entropy increases irreversibly along the flow path due to frictional dissipation, distinguishing Fanno flow from isentropic processes.[1] Fanno flow has practical applications in engineering, such as analyzing gas transport in pipelines, designing rocket engine nozzles where frictional choking limits performance, and modeling internal flows in propulsion systems.[2] Numerical tools like the Generalized Fluid System Simulation Program (GFSSP) validate these models by accurately predicting property distributions under frictional constraints.[2]Fundamentals
Definition and Assumptions
Fanno flow refers to the steady, one-dimensional, compressible flow of an ideal gas through a duct of constant cross-sectional area, where wall friction effects are significant but there is no heat transfer or work done on the fluid.[1] This model captures the behavior of gases in pipelines or nozzles under frictional adiabatic conditions, distinguishing it from frictionless isentropic flows by incorporating irreversible losses due to shear at the walls.[1] The concept is named after Gino Girolamo Fanno, an Italian mechanical engineer who first analyzed it in his 1904 master's thesis on compressible gas flow in ducts.[4] Fanno's work laid the foundation for understanding how friction alters flow properties in constant-area conduits, influencing modern applications in propulsion and piping systems. Key assumptions underlying the Fanno flow model include a steady-state process, one-dimensional flow approximation where velocity varies only along the duct axis, and treatment of the fluid as an ideal gas with constant specific heats.[1] The flow is adiabatic, meaning no heat transfer occurs (q = 0), and the duct has constant cross-sectional area (dA = 0), with negligible body forces such as gravity.[1] Friction is modeled using the Darcy-Weisbach equation with a constant friction factor f, assuming fully developed turbulent flow and ignoring entrance effects.[5] Initial conditions for Fanno flow typically involve subsonic (Mach number M < 1) or supersonic (M > 1) inlet flow, with the duct length determining whether the flow reaches sonic conditions (M = 1) at the exit, known as choking, if friction is sufficient to decelerate or accelerate the flow to the speed of sound.[1] Boundary conditions are set by specifying inlet stagnation properties and the friction parameter, ensuring the flow remains choked for maximum mass flow rate under given constraints.[1]Governing Equations
The governing equations for Fanno flow are derived from the fundamental conservation laws applied to one-dimensional, steady, compressible flow in a constant-area duct with wall friction and no heat transfer or external work. These equations describe the evolution of flow properties along the duct length.[6] The continuity equation, expressing mass conservation, states that the mass flow rate is constant: \rho u A = \constant For a duct of constant cross-sectional area A, this simplifies to the differential form: \frac{d\rho}{\rho} + \frac{du}{u} = 0 where \rho is the fluid density and u is the flow velocity.[1][6] The momentum equation accounts for the balance between pressure forces, inertial effects, and frictional shear at the wall. In differential form, it is: \rho u \, du + dp + \left( \frac{f \, dx}{D_h} \right) \frac{\rho u^2}{2} = 0 where p is the static pressure, f is the Darcy friction factor (a dimensionless measure of wall shear stress), and D_h is the hydraulic diameter of the duct. The term \frac{f \, dx}{D_h} \frac{\rho u^2}{2} represents the frictional pressure drop over an infinitesimal length dx.[6][3] The energy equation reflects the adiabatic nature of the flow, with total enthalpy conserved along a streamline: h + \frac{u^2}{2} = \constant For an ideal gas with constant specific heat, this becomes: T + \frac{u^2}{2 c_p} = \constant where h = c_p T is the static enthalpy, T is the static temperature, and c_p is the specific heat at constant pressure.[1][6] The equation of state for an ideal gas closes the system of equations: p = \rho R T where R is the specific gas constant.[3][6] A key non-dimensional parameter in Fanno flow analysis is the Mach number, defined as: M = \frac{u}{a} where a = \sqrt{\gamma R T} is the local speed of sound and \gamma = c_p / c_v is the ratio of specific heats. The Mach number characterizes the flow regime, with subsonic (M < 1) and supersonic (M > 1) behaviors differing significantly due to friction.[1][3]Theoretical Derivation
Conservation Laws Application
The application of conservation laws to Fanno flow involves deriving differential equations that describe the evolution of flow properties along the duct length, based on steady, one-dimensional, adiabatic flow of an ideal gas in a constant-area conduit with wall friction. The continuity equation ensures constant mass flux, \rho u = \dot{m}/A = constant, leading to \frac{d\rho}{\rho} + \frac{du}{u} = 0. The energy equation, reflecting adiabatic conditions with no shaft work or heat transfer, conserves total enthalpy: h + \frac{u^2}{2} = constant. For an ideal gas, this simplifies to c_p T + \frac{u^2}{2} = constant, yielding the differential relation \frac{dT}{T} = -(\gamma - 1) M^2 \frac{du}{u}, where \gamma is the specific heat ratio and M is the Mach number.[7] The momentum equation accounts for pressure forces, convective momentum changes, and frictional shear at the wall. For a differential control volume of length dx, the balance gives dp \, A + \rho u \, du \, A = -\tau_w \, P \, dx, where A is the cross-sectional area, P is the wetted perimeter, and \tau_w = f \frac{\rho u^2}{2} is the wall shear stress using the Fanning friction factor f. Substituting and nondimensionalizing by \rho u^2 yields \frac{dp}{\rho u^2} + \frac{du}{u} + \frac{4 f \, dx}{D_h} = 0, where D_h = 4 A / P is the hydraulic diameter. Since M^2 = u^2 / a^2 = \rho u^2 / (\gamma p) with speed of sound a = \sqrt{\gamma p / \rho}, this becomes \frac{dp}{p} + \gamma M^2 \frac{du}{u} + \gamma M^2 \left( \frac{4 f \, dx}{D_h} \right) = 0.[7][8] Combining these with the ideal gas law p = \rho R T (so \frac{dp}{p} = \frac{d\rho}{\rho} + \frac{dT}{T}) eliminates intermediate variables like pressure and temperature to obtain a single ordinary differential equation governing the Mach number evolution. Substituting the continuity and energy relations into the momentum equation gives \frac{dM^2}{M^2} = \frac{\gamma M^2 \left(1 + \frac{\gamma - 1}{2} M^2 \right)}{1 - M^2} \frac{4 f \, dx}{D_h}, where D_h is the hydraulic diameter; this form serves as the precursor to the integrated Fanno parameter \frac{4 f L^*}{D_h}. For subsonic inlet flow (M < 1), friction accelerates the flow (du > 0), increasing M toward unity; for supersonic inlet (M > 1), it decelerates toward unity. Friction also leads to an increase in entropy along the duct, as required by the second law of thermodynamics for irreversible processes.[7][8] A key outcome is the choking condition: as M approaches 1, the denominator $1 - M^2 approaches zero, making \frac{dM^2}{dx} infinite, which prevents further acceleration or deceleration without external adjustments. Thus, sonic conditions (M = 1) occur at a finite duct length L^* from the inlet, establishing the maximum achievable mass flow rate for given stagnation conditions and duct geometry, beyond which the flow chokes and upstream conditions must adjust to maintain continuity.[7]Normalization and Non-Dimensional Parameters
To analyze Fanno flow independently of specific inlet conditions or duct geometries, the governing equations are normalized using non-dimensional parameters that scale properties relative to a reference state at the sonic choking point. This reference state, denoted by an asterisk (*), corresponds to the location where the flow reaches Mach number M^* = 1, with associated sonic pressure p^*, temperature T^*, density \rho^*, and velocity u^* = a^* (the speed of sound at sonic conditions). The choking point represents the maximum length for a given inlet Mach number before the flow becomes sonic, beyond which adjustments in upstream conditions are required to maintain continuity.[7][3] The primary non-dimensional parameter for characterizing frictional effects in Fanno flow is the length scale \frac{4f L^*}{D_h}, where f is the Fanning friction factor, L^* is the distance from the inlet to the sonic reference point, and D_h is the hydraulic diameter of the duct. This parameter arises from integrating the differential momentum equation along the duct, which relates changes in Mach number to friction and geometry. The derivation begins with the continuity equation (\rho u A = \dot{m} = constant), the energy equation (constant stagnation temperature T_0), and the momentum equation incorporating wall shear stress \tau_w = f \frac{\rho u^2}{2}, leading to the differential form \frac{dM}{dx} = \frac{\gamma M}{2(1 + \frac{\gamma-1}{2} M^2)} \cdot \frac{4f}{D_h} \cdot \frac{1 + \gamma M^2}{1 - M^2}. Rearranging and integrating from an initial Mach number M to the sonic condition M = 1 yields the non-dimensional length as \frac{4f L^*}{D_h} = \int_M^1 \frac{1 - M'^2}{\gamma M'^4 \left(1 + \frac{\gamma-1}{2} M'^2 \right)} d(M'^2). The closed-form solution for this integral, obtained through algebraic manipulation and logarithmic integration, is \frac{4f L^*}{D_h} = \frac{1 - M^2}{\gamma M^2} + \frac{\gamma + 1}{2\gamma} \ln \left( \frac{(\gamma + 1) M^2}{2 + (\gamma - 1) M^2} \right). This expression allows direct computation of the choking length for any inlet M < 1 (subsonic) or M > 1 (supersonic), with the Mach number serving as the key independent variable for tabulating flow behavior.[7][3][1] Another important non-dimensional quantity in Fanno flow analysis is the impulse function, defined as I = p A (1 + \gamma M^2), where A is the constant duct cross-sectional area. This function emerges from the integrated momentum equation, balancing pressure forces, momentum flux, and frictional losses: dI = -4 \tau_w dx, indicating that I decreases monotonically along the duct due to irreversible friction, even as stagnation enthalpy remains constant. The variation of I relative to its sonic value I^* = p^* A (1 + \gamma) provides insight into thrust or force requirements for ducted flows, with I / I^* = \frac{p / p^* (1 + \gamma M^2)}{1 + \gamma}.[9][3]Flow Characteristics
Thermodynamic Property Variations
In Fanno flow, the variations in thermodynamic properties along the constant-area duct are governed by the effects of wall friction under adiabatic conditions, with total temperature remaining constant. These variations are expressed in non-dimensional form relative to the sonic reference state (denoted by *), where the Mach number M = 1 at the choking point. In the subsonic regime (M < 1), as the flow progresses downstream toward choking, the Mach number increases, the static velocity increases, the static temperature and static pressure decrease, the density decreases, and the stagnation pressure decreases due to frictional irreversibilities. In the supersonic regime (M > 1), as the flow progresses downstream toward choking, the Mach number decreases, the static velocity decreases, the static temperature and static pressure increase, the density increases, and the stagnation pressure decreases.[10] The increase in specific entropy along the duct in both regimes reflects the irreversible nature of friction, given by the differential relation ds = c_p \frac{dT}{T} - R \frac{dp}{p} > 0, where the pressure drop term dominates, ensuring positive entropy generation despite the temperature change.[10] The key non-dimensional ratios for an ideal gas with constant specific heat ratio γ are derived from the conservation laws and are as follows. The static temperature ratio is \frac{T}{T^*} = \frac{\gamma + 1}{2 + (\gamma - 1) M^2}. The static velocity ratio is \frac{u}{u^*} = M \sqrt{\frac{T}{T^*}} = M \sqrt{ \frac{\gamma + 1}{2 + (\gamma - 1) M^2} }. The static pressure ratio is \frac{p}{p^*} = \frac{1}{M} \sqrt{ \frac{T}{T^*} } = \frac{1}{M} \sqrt{ \frac{\gamma + 1}{2 + (\gamma - 1) M^2} }. The stagnation pressure ratio is \frac{p_0}{p_0^*} = \frac{p}{p^*} \left( \frac{2 + (\gamma - 1) M^2}{\gamma + 1} \right)^{\frac{\gamma}{\gamma - 1}} = \frac{1}{M} \left( \frac{2 + (\gamma - 1) M^2}{\gamma + 1} \right)^{\frac{\gamma + 1}{2(\gamma - 1)}}. These relations hold for both subsonic and supersonic branches, with the reference sonic state representing the endpoint if the duct were sufficiently long to reach choking.[10][7] For air modeled as an ideal gas with γ = 1.4, the following table illustrates representative property ratios at selected Mach numbers, computed using the above equations to highlight the trends in each regime.| M | T/T* | p/p* | u/u* | p₀/p₀* |
|---|---|---|---|---|
| 0.2 | 1.1905 | 5.455 | 0.2182 | 2.9635 |
| 0.5 | 1.1429 | 2.138 | 0.5345 | 1.340 |
| 0.8 | 1.0638 | 1.289 | 0.825 | 1.038 |
| 1.0 | 1.0000 | 1.000 | 1.000 | 1.000 |
| 1.2 | 0.9315 | 0.805 | 1.159 | 1.031 |
| 1.5 | 0.8276 | 0.607 | 1.365 | 1.177 |
| 2.0 | 0.6667 | 0.408 | 1.633 | 1.686 |
| 3.0 | 0.4286 | 0.218 | 1.964 | 4.235 |