Sound propagation in narrow tubes including effects of viscothermal and turbulent damping with application to charge air coolers

Charge air coolers (CACs) are used on turbocharged internal combustion engines to enhance the overall gas-exchange performance. The cooling of the charged air results in higher density and thus volumetric efficiency. It is also important for petrol engines that the knock margin increases with reduced charge air temperature. A property that is still not very well investigated is the sound transmission through a CAC. The losses, due to viscous and thermal boundary layers as well as turbulence, in the narrow cooling tubes result in frequency dependent attenuation of the transmitted sound that is significant and dependent on the flow conditions. Normally, the cross-sections of the cooling tubes are neither circular nor rectangular, which is why no analytical solution accounting for a superimposed mean flow exists. The cross-dimensions of the connecting tanks, located on each side of the cooling tubes, are large compared to the diameters of the inlet and outlet ducts. Three-dimensional effects will therefore be important at frequencies significantly lower than the cut-on frequencies of the inlet/outlet ducts. In this study the two-dimensional finite element solution scheme for sound propagation in narrow tubes, including the effect of viscous and thermal boundary layers, originally derived by Astley and Cummings [Wave propagation in catalytic converters: Formulation of the problem and finite element scheme, Journal of Sound and Vibration 188 (5) (1995) 635–657] is used to extract two-ports to represent the cooling tubes. The approximate solutions for sound propagation, accounting for viscothermal and turbulent boundary layers derived by Dokumaci [Sound transmission in narrow pipes with superimposed uniform mean flow and acoustic modelling of automobile catalytic converters, Journal of Sound and Vibration 182 (5) (1995) 799–808] and Howe [The damping of sound by wall turbulent shear layers, Journal of the Acoustical Society of America 98 (3) (1995) 1723–1730], are additionally calculated for corresponding circular cross-sections for comparison and discussion. The two-ports are thereafter combined with numerically obtained multi-ports, representing the connecting tanks, in order to obtain the transmission properties for the charged air when passing the complete CAC. An attractive formalism for representation of the multi-ports based on the admittance relationship between the ports is presented. From this the first linear frequency domain model for CACs, which includes a complete treatment of losses in the cooling tubes and 3D effects in the connecting tanks is extracted in the form of a two-port. The frequency dependent transmission loss is calculated and compared to the corresponding experimental data with good agreement.