Turbomachinery Flow Physics and Dynamic Performance

Turbomachines are devices within which conversion of total energy of a working medium into mechanical energy and vice versa takes place. Turbomachines are generally divided into two main categories. The first category is used primarily to produce power. It includes, among others, steam turbines, gas turbines, and hydraulic turbines. The main function of the second category is to increase the total pressure of the working fluid by consuming power. It includes compressors, pumps, and fans.

The turbomachinery aerodynamic design process has been experiencing continuous progress using the Computational Fluid Dynamics (CFD) tools. The use of CFD-tools opens a new perspective in simulating the complex three-dimensional (3-D) turbomachinery flow field. Understanding the details of the flow motion and the interpretation of the numerical results require a thorough comprehension of fluid mechanics laws and the kinematics of fluid motion within the turbomachinery component. Kinematics is treated in many fluid mechanics texts. Aris [1] and Spurk [2] give excellent accounts of the subject. In the following sections, a compact and illustrative treatment is given to cover the basic needs of the reader.

In this and the following chapter, we present the conservation laws of fluid mechanics that are necessary to understand the basics of flow physics in turbomachinery from a unified point of view. The main subject of this chapter is the differential treatment of the conservation laws of fluid mechanics, namely conservation law of mass, linear momentum, angular momentum, and energy. These subjects are treated comprehensively in a recent book by Schobeiri [1]. In many engineering applications, such as in turbomachinery, the fluid particles change the frame of reference from a stationary frame followed by a rotating one. The absolute frame of reference is rigidly connected with the stationary parts, such as casings, inlets, and exits of a turbine, a compressor, a stationary gas turbine or a jet engine, whereas the relative frame is attached to the rotating shaft, thereby turning with certain angular velocity about the machine axis. By changing the frame of reference from an absolute frame to a relative one, certain flow quantities remain unchanged, such as normal stress tensor, shear stress tensor, and deformation tensor. These quantities are indifferent with regard to a change of frame of reference. However, there are other quantities that undergo changes when moving from a stationary frame to a rotating one. Velocity, acceleration, and rotation tensor are a few. We first apply these laws to the stationary or absolute frame of reference, then to the rotating one.

The differential analysis is of primary significance to all engineering applications such as compressor, turbine, combustion chamber, inlet, and exit diffuser, where a detailed knowledge of flow quantities, such as velocity, pressure, temperature, entropy, and force distributions, are required. A complete set of independent conservation laws exhibits a system of partial differential equations that describes the motion of a fluid particle. Once this differential equation system is defined, its solution delivers the detailed information about the flow quantities within the computational domain with given initial and boundary conditions.

In the following sections, we summarize the conservation laws in integral form essential for applying to the turbomachinery flow situations. Using the Reynolds transport theorem explained in Chapter 2, we start with the continuity equation, which will be followed by the equation of linear momentum, angular momentum, and the energy. Vavra [1] utilized an alternative approach by directly integrating the differential balances. Both approaches are valid and lead to the same results.

The energy transfer in turbomachinery is established by means of the stages. A turbomachinery stage comprises a row of fixed, guide vanes called stator blades, and a row of rotating blades termed rotor. To elevate the total pressure of a working fluid, compressor stages are used that partially converts the mechanical energy into potential energy. According to the conservation law of energy, this energy increase requires an external energy input which must be added to the system in the form of mechanical energy. Figure 5.1 shows the schematic of an axial compressor stage that consists of one stator and two rotor rows. In general, a compressor component starts with a rotor row followed by a stator row. Compressor configurations are also found that starts with an inlet guide vane. To define a unified station nomenclature for compressor and turbine stages, we identify with station number 1 as the inlet of the stator, followed by station 2 as the rotor inlet, and 3, rotor exit. The absolute and relative flow angles are counted counterclockwise from a horizontal line. This convention allows an easier calculation of the off-design behavior of compressor and turbine stages during a transient operation, as we will see later. This angle definition is different the angle conventions used in literature, [1], [2], [3], and [4].

The last chapter was dedicated to the energy transfer within turbomachinery stages. The stage mechanical energy production or consumption in turbines and compressors were treated from a unified point of view by introducing a set of dimensionless parameters. As shown in Chapter 4, the mechanical energy, and therefore the stage power, is the result of the scalar product between the moment of momentum acting on the rotor and the angular velocity. The moment of momentum in turn was brought about by the forces acting on rotor blades. The blade forces are obtained by applying the conservation equation of linear momentum to the turbine or compressor cascade under investigation. In this chapter, we first assume an inviscid flow for which we establish the relationship between the lift force and circulation. Then, we consider the viscosity effect that causes friction or drag forces on the blading.

In Chapter 7, we derived the equations for calculation of different losses that occur within a stage of a turbomachine. As shown, the sum of those losses determines the stage efficiency which shows the capability of energy conversion within the stage. The stage efficiency, however, is not fully identical with the efficiency of the entire turbomachine. In a multi-stage turbomachine, the expansion or compression process within individual stages causes an entropy production which is associated with a temperature increase. For an expansion process, this temperature increase leads to a heat recovery and for a compression process, it is associated with a reheat ash shown in Fig. 8.1. As a consequence, the turbine efficiency is higher and the compressor efficiency lower than the stage efficiency. The objective of this chapter is to describe this phenomenon by means of classical thermodynamic relations. The approach is adopted by many authors, among others, Traupel [1] and Vavra [2].

Up to this point, the relationships developed for a turbomachinery stage have been strictly correct for given velocity diagrams with known inlet and exit flow angles. We assumed that the flow is fully congruent with the blade profile. This assumption implies that the inlet and exit flow angles coincide with the camber angles at the leading and trailing edges. Based on the operation condition and the design philosophy, there might be a difference between the camber and flow angle at the leading edge, which is called the incidence angle. The difference between the blade camber angle and the flow angle at the exit is termed the deviation angle. Since the incidence and deviation affect the required total flow deflection, the velocity diagram changes. If this change is not predicted accurately, the stage operates under a condition not identical with the optimum operation condition for which the stage is designed. This situation affects the efficiency and performance of the stage and thus the entire turbomachine. In order to prevent this, the total flow deflection must be accurately predicted. The compressor and the turbine flows react differently to a change of incidence. For instance, a slight change of incidence causes a partial flow separation on the compressor blade suction surface that can trigger a rotating stall; a turbine blade is less sensitive even to greater incidence change. To obtain the incidence and deviation angle for compressor and turbine blades, we use two different calculation methods. The first method deals with the application of conformal transformation to cascade flows with low deflection as in compressor blades. The second method concerns the calculation of deviation in high loaded cascades as in turbine blades.

Flow deflection in turbomachines is established by stator and rotor blades with prescribed geometry that includes inlet and exit camber angles, stagger angle, camber line, and thickness distribution. The blade geometry is adjusted to the stage velocity diagram which is designed for specific turbine or compressor flow applications. Simple blade design methods are available in the open literature (see References). More sophisticated and high efficiency blade designs developed by engine manufacturers are generally not available to the public. An earlier theoretical approach by Joukowsky [1] uses the method of conformal transformation to obtain cambered profiles that can generate lift force. The mathematical limitations of the conformal transformation do not allow modifications of a cambered profile to produce the desired pressure distribution required by a turbine or a compressor blade design. In the following, a simple method is presented that is equally applicable for designing compressor and turbine blades. The method is based on (a) constructing the blade camber line and (b) superimposing a predefined base profile on the camber line. With regard to generating a base profile, the conformal transformation can be used to produce useful profiles for superposition purposes. A brief description of the Joukowsky transformation explains the methodology of symmetric and a-symmetric (Cambered) profiles. The transformation uses the complex analysis which is a powerful tool to deal with the potential theory in general and the potential flow in particular. It is found in almost every fluid mechanics textbook that has a chapter dealing with potential flow. While they all share the same underlying mathematics, the style of describing the subject to engineering students differ. A very compact and precise description of this subject matter is found in an excellent textbook by Spurk [2].

In Chapter 5, we briefly described a simple radial equilibrium condition necessary to determine the radial distribution of the stage parameters such as, φ, λ, r, α _i , and β _i . Assuming an axisymmetric flow with constant meridional velocity and total pressure distributions, we arrived at free vortex flow as a simple radial equilibrium condition with \(r\nu_u = \mbox{const}\). In practice, from an aerodynamics design point of view, a constant meridional velocity component or constant total pressure may not be desirable. As an example, consider the flow field close to the hub or tip of a stage where secondary flow vortices predominate. As discussed in Chapter 6, these secondary vortices induce drag forces leading to secondary flow losses which reduce the efficiency of the stage. To reduce the secondary flow losses, specific measures can be taken that are not compatible with the simple radial equilibrium condition. In this case, the simple radial equilibrium method needs to be replaced by a general one. Wu [1] proposed a general theory for calculating the three-dimensional flow in turbomachines. He introduced two sets of surfaces: Blade to blade surfaces called S_1i and hub-to-tip surfaces labeled with S_2j . Utilizing S_1i and S_2j surfaces, Wu [1] proposed an iterative method to solve the three-dimensional inviscid flow field in turbomachinery stages. Coupling both surfaces, however, is associated with computational instabilities that gives rise to replacing the technique by complete Euler or Navier-Stokes solver, [2]. A computationally more stable method for solving the flow field is the streamline curvature technique. This method is widely used in turbomachinery industry and is the essential tool for generating the basic design structure necessary to start with CFD application. The streamline curvature method can be used for design, off-design, and analysis. Vavra [3] presented the theoretical structure for inviscid axisymmetric flow in turbomachines that can be used to derive the streamline curvature equations. A thorough review of the streamline curvature method can be found in Novak and Hearsey [4], Wilkinson [5], and Lakshminarayana [2]. Wennerstrom [6] presented a concise description of this technique which is given in this section in its original form. Rapid calculation procedures used in the turbomachinery industry determining the distribution of flow properties within the turbomachinery assume steady adiabatic flow and axial symmetry. The more sophisticated of these procedures include calculation stations within blade rows as well as the more easily treated stations at the blading, leading and trailing edges. The assumption of axial symmetry in the bladed region implies an infinite number of blades. The blade forces acting between the blades and the fluid are taken into consideration by body force terms in equation of motion. The streamlines are not straight lines as could be supposed. They usually have certain curvatures that are maintained if the forces exerted on the fluid particle are in an equilibrium state described by the equilibrium equation. This equation includes various derivatives in the meridional plane and is solved in this plane along the computing stations which are normal to the average meridional flow direction. The equilibrium equation in its complete form cannot be solved analytically, therefore, numerical calculation methods are necessary. For the application of the equilibrium equation to the stage flow with streamline curvature, Hearsey [7] developed a comprehensive computer program which is successfully used in the turbomachinery industry for design of advanced compressors and turbines. Since the treatment of the corresponding numerical method for the solution of this equation is given in [7], this chapter discusses the basic physical description of the calculation method. As a result, the streamline curvature method can be used as a design tool and for the post design analysis. An advanced compressor design is presented followed by a brief discussion of special cases.

The following chapters deal with the nonlinear transient simulation of turbomachinery systems. Power generation steam and gas turbine engines, combined cycle systems, aero gas turbine engines ranging from single spool engines to multi-spool high pressure core engines with an afterburner for supersonic flights, rocket propulsion systems and compression systems for transport of natural gas with a network of pipeline systems are a few examples of systems that heavily involve turbomachinery components.

Considering a power generation gas turbine engine as a turbomachinery system that is designed for steady state operation, its behavior during routine startups, shot downs, and operational load changes significantly deviates from the steady state design point. Aero gas turbine engines have to cover a relatively broad operational envelope that includes takeoff, low and high altitude operation conditions, as well as landing. During these operations, the components are in a continuous dynamic interaction with each other, where the aero-thermodynamic as well as the mechanical load conditions undergo temporal changes. As an example, the acceleration/ deceleration process causes a dynamic mismatch between the turbine and compressor power resulting in temporal change of the shaft speed.

In the above cases, the turbomachinery systems are subjected to the operating modes that are specific to the system operation. Besides these foreseeable events, there are unforeseeable operation scenarios that are not accounted for when designing the system. System failures, such as, blade loss during a routine operation, loss of cooling mass flow through the cooled turbine blades, adverse operation conditions that force the compressor component to surge, and failure of the control system, are a few examples of adverse operational conditions. In all of these operations, the system experiences adverse changes in total fluid and the thermodynamic process leading to greater aerodynamic, thermal and mechanical stress conditions.

The trend in the development of gas turbine technology during the past decades shows a continuous increase in efficiency, performance, and specific load capacities. This trend is inherently associated with increased aerodynamic, thermal, and mechanical stresses. Under this circumstance, each component operates in the vicinity of its aerodynamic, thermal and mechanical stress limits. Adverse operational conditions that cause a component to operate beyond its limits can cause structural damages as a result of increased aerodynamic, thermal, and structural stresses. To prevent this, the total response of the system, including aerodynamic, thermal, and mechanical responses must be known in the stage of design and development of new turbomachinery systems.

This chapter describes the physical basis for the non-linear dynamic simulation of gas turbine components and systems. A brief explanation of the numerical method for solution is followed by detailed dynamic simulation of several components described in the following chapters.

A turbomachinery system such as a power generation gas turbine engine, a thrust generation aero- engine, rocket propulsion, or a small turbocharger, consist of several sub-systems that we call components ([1],[2],[3],[4]). Each component is an autonomous entity with a defined function within the system. Inlet nozzles, exit diffusers, combustion chambers, compressors, and turbines are a few component examples. A component may consist of several sub-components. A turbine or a compressor stage exhibits such a sub-component. The numerical models of components are called modules.

This chapter deals with the numerical modeling of the components pertaining to group 1 discussed in section 13.1.1 The components pertaining to this category are the connecting pipes, inlet and exhaust systems, as shown in Fig. 14.1.

This category of components includes recuperators, preheaters, regenerators, intercoolers, and aftercoolers. Within these components the process of heat exchange occurs between the high and low temperature sides. The working principle of these components is the same ([1], [2], [3]). However, different working media are involved in the heat transfer process. More recently recuperators are applied to small and medium size gas turbine engines to improve their thermal efficiency. The exhaust thermal energy is used to warm up the compressor exit air before it enters the combustion chamber. A typical recuperator consists of a low pressure hot side flow path, a high pressure cold side flow path, and the wall that separates the two flow paths. A variety of design concepts are used to maximize the heat exchange between the hot and the cold side by improving the heat transfer coefficients. A cold side flow path may consist of a number of tubes with turbulators, fin pins, and other features that enhance the heat transfer coefficient. Based on the individual recuperator design concept, hot gas impinges on the tube surface in cross flow or counter flow directions. The working media entering and exiting the recuperator is generally combustion gas that exits the diffuser (hot side) and air that exits the compressor (cold side).

As mentioned in Chapter 1, the function of a compressor is to increase the total pressure of the working fluid. According to the conservation law of energy, this total pressure increase requires external energy input, which must be added to the system in the form of mechanical energy. The compressor rotor blades exert forces on the working medium thereby increasing its total pressure. Based on efficiency and performance requirements, three types of compressor designs are applied. These are axial flow compressors, radial or centrifugal compressors, and mixed flow compressors. Axial flow compressors are characterized by a negligible change of the radius along the streamline in the axial direction. As a result, comparison of the contribution of the circumferential kinetic energy difference \((U^2_3 - U^2_2)/2\) to the pressure buildup is marginal. In contrast, the above difference is substantial for a radial compressor stage, where it significantly contributes to increasing the total pressure as discussed in Chapter 5.

During the compression process, the fluid particles are subjected to a positive pressure gradient environment that may cause the boundary layer along the compressor blade surface to separate. To avoid separation, the flow deflection across each stage and thus the stage pressure ratio is kept within certain limits discussed in the following section. Compared to an axial stage, much higher relative stage pressure ratios π _rad /π _ax  > 5 at relatively moderate flow deflections can be achieved by centrifugal compressors. However, geometry, mass flow, efficiency, and material constraints place limits on utilizing radial compressors. Radial compressors designed for high stage pressure ratios and mass flows comparable to those of axial compressors require substantially larger exit diameters. This can be considered an acceptable solution for industrial applications, but is not a practical solution for implementing into gas turbine engines. In addition, for gas turbine applications, high compressor efficiencies are required to achieve acceptable thermal efficiencies. While the die efficiencies of advanced axial compressors have already exceeded 91.5% range, those of advanced centrifugal compressors are still below 90%. Power generation gas turbine engines of 10 MW and above as well as medium and large aircraft engines use axial compressor design. Small gas turbines, turbochargers for small and large Diesel engines have radial impellers that generate pressure ratios above 5. Compact engines for turboprop applications may have a combination of both. In this case a relatively high efficiency multi-stage axial compressor is followed by a lower efficiency centrifugal compressor to achieve the required engine pressure ratio at smaller stage numbers.

Further stage pressure buildup is achieved by increasing the inlet relative Mach number M _2rel. = W ₂/c ₂. In case of subsonic axial flow compressors with M _2rel. < 1, the compression process is primarily established by diffusion and flow deflection. However, in the case of supersonic relative Mach number M _2rel. > 1 that occupies the entire compressor blade height from hub to tip, the formation of oblique shock waves followed by normal shocks as discussed in Chapter 4 substantially contributes to a major pressure increase. However, the increase of stage pressure ratio as a result of compression shocks is associated with additional shock losses that reduce the stage efficiency. To achieve a higher stage pressure ratio at an acceptable loss level, the compressor stage can be designed as transonic compressor stage. In this case, the relative Mach number at the hub is subsonic and at the tip supersonic, with transonic Mach range in between. Transonic compressor stage design is applied to the first compressor stage with a relatively low aspect ratio of high performance gas turbine engines.

In this chapter, we first investigate several loss mechanisms and correlations that are specific to compressor component. Using these correlations, first the basic concept for a row-by-row adiabatic calculation method is presented that accurately predicts the design and off-design behavior of single and multi-stage compressors. With the aid of this method, efficiency and performance maps are easily generated. The chapter is then enhanced to calculate the diabatic compression process where the blade rows exchange thermal energy with the working medium and vice versa. The above methods provide three different options for dynamically simulating the compressor component. The first option is to utilize the steady state compressor performance maps associated with dynamic coupling. The second option considers the row-by-row adiabatic calculation. Finally, the third option uses the diabatic compression process. Examples are presented.

A gas turbine engine is a system that consists of several turbomachinery components and auxiliary subsystems. Air enters the compressor component which is driven by a turbine component and is placed on the same shaft. Air exits the compressor at a higher pressure and enters the combustion chamber, where the chemical energy of the fuel is converted into thermal energy producing combustion gas at a temperature that corresponds to the turbine inlet design temperature. The combustion gas expands in the following turbine component, where its total energy is partially converted into shaft work and exit kinetic energy. For power generation gas turbines, the shaft work is the major portion of the above energy forms. It covers the total work required by the compressor component, the bearing frictions, several auxiliary subsystems, and the generator. In aircraft gas turbines, a major portion of the total energy goes toward generation of high exit kinetic energy that is essential for thrust generation.

Considering the flows within the blade boundary layer, based on the blade geometry and pressure gradient, three distinctly different flow patterns can be identified: 1) laminar flow (or non-turbulent flow) characterized by the absence of stochastic motions, 2) turbulent flow, where flow pattern is determined by a fully stochastic motion of fluid particles, and 3) transitional flow characterized by intermittently switching from laminar to turbulent at the same spatial position. Of the three patterns, that is predominant in turbomchinery components is the transitional flow pattern. The Navier-Stokes equations presented in Chapter 3 generally describe the unsteady flow through a turbomachinery component. Using a direct numerical simulation (DNS) approach delivers the most accurate results. However, the application of DNS, for the time being, is restricted to simple flows at low Reynolds numbers. For calculating the complex turbomachinery flow field within a reasonable time frame, Reynolds averaged Navier-Stokes (RANS) equations with appropriate turbulence models are used. A detailed discussion of this topic is found in Chapter 20.

The preceding Chapter dealt with stability of laminar flows, their perturbation and transition to the turbulent state. In discussing the transition process, we prepared the essentials for better understanding the basic physics of the more complex turbulent flow, which is still an unresolved and extremely challenging problem in fluid mechanics.

In Chapter 19 we have shown that using the computational fluid dynamics (CFD), flow details in and around complex geometries can be predicted with an acceptable degree of accuracy. The flow field calculation includes details very close to the wall, where the viscosity plays a significant role. In the absence of random fluctuations the (laminar) flow can be calculated with high accuracy. For predicting turbulent flows, however, turbulence models were required to be implemented into the Navier-Stokes equations to account for turbulence fluctuations. One of the more important tasks in turbomachinery fluid mechanics is to predict the drag forces acting on the surfaces such as turbine and compressor blade surfaces, endwalls, inlet nozzles and exit diffusers. As seen in Chapter 6, the drag forces are produced by the fluid viscosity which causes the shear stress acting on the surface. The question that arises is how far from the surface the viscosity dominates the flow field. Prandtl [1] was the first to answer this question. Combining his physical intuition with experiments, he developed the concept of the boundary layer theory. This theory in its differential or integral forms can be applied to a turbomachinery flow to determine the blade profile loss. It delivers accurate results as long as the boundary layer is not separated.